Kurdistan Regional Government Ministry of Higher Education and Scientific Research University of Sulaimani College of Education Department of Mathematics

Some Properties of Certain Types of Modules That are Preserved under Localization A Thesis Submitted to the Council of College of Education at the University of Sulaimani in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics (Algebra) By

Bakhtyar Mahmood Rahim B. Sc. Mathematics (2012), University of Sulaimani Supervised By

Prof. Dr. Adil K. Jabbar

July-2017

‫‪‬‬ ‫وَفَوْقَ كُلِّ ذِى عِلْمٍ عَلِيمٌ‬ ‫‪‬‬ ‫سورة يوسف‪ ،‬آية (‪)76‬‬

DEDICATION

This thesis is dedicated to:  The memory of My Parents, My wife;  My brothers and sisters;  All my friends.

With love and Respect

ACKNOWLEDGMENTS First of all, I would like to thank the God for everything. I would like to express my deepest gratitude to my supervisor, Professor Dr. Adil K. Jabbar, for his support, guidance, encouragement, recommendation to accomplish this thesis. Without his amazing experience, the fieldwork would never have been carried out successfully. I would like to extend my gratitude and thanks to the dean and the staff members of the College of Education, especially the staff of Mathematics Department. A special thanks goes to my family. Words cannot express how grateful I am to my parents, my brothers and sisters for their support and encouragement. I would also like to thank all of my friends and all of the people who supported me in writing, and made me strive towards my goal.

Abstract

Abstract In this work, some algebraic structures are studied, such as small submodules, supplement submodules, semi simple submodules, w-supplemented modules, lie above submodules, amply weak supplemented modules, amply cofinitely weak supplemented modules, Rad-supplemented modules, weak Rad-supplemented modules and several results that concern these structures are proved. Furthermore, some relations that combine these types of modules each with the other are determined. The main aim of this work is to determine those properties of some types of modules that are preserved under localization. Also, the work aims at extending some known properties of modules under effect of localization. In addition to the above, several conditions are obtained that make the localization of an amply weak supplemented module as an amply weak supplemented module and some other conditions are given each of which makes some properties of certain types of modules can be transformed from the localization of given modules to the modules itself, we prove that under certain condition the amply weak supplementedness property of modules can be transformed from the localization of a given module to the module itself. Finally, further conditions are given under which certain properties of 𝑅𝑎𝑑 −supplemented modules and weak 𝑅𝑎𝑑 −supplemented modules are preserved under localization.

I

Contents

Contents Abstract………………………………………………………….………………..I Contents ………………………………………………………….………………II List of Abbreviations …………………..……………………….………………III Introduction………………………………………………………………............1

Chapter One: The Fundamentals 1.1 The Construction of 𝑅𝑆 and 𝑀𝑆 ………………………………………………..3 1.2 Some Definitions and examples……………………………………………….5 1.3 Some Known Results ……………………………………………………….10

Chapter Two: Amply Weak Supplemented Modules and Rad- Supplemented Modules 2.1 Amply Weak Supplemented Modules……………………………………...…24 2.2 Rad- Supplemented Modules…………………………………….....…………35

Chapter Three: Small Submodule, Semi Simple Submodule and Some Types of Supplemented Modules 3.1 Small and Semi Simple Submodule ……………………………………….…48 3.2 Weak Rad- and Some others Types of Supplemented Modules…………...…53 References ……………………………………………………………...…...........63

II

List of Abbreviations

List of Abbreviations Symbols

Descriptions

𝐾≤𝑀

𝐾 is a submodule of the module 𝑀

𝐾≪𝑀

𝐾 is a small submodule of the module 𝑀

Rad (𝑀) 𝑈⨁𝑉

the radical of the module 𝑀 direct sum of two submodules of the module 𝑀

𝑅𝑆

localization of the ring 𝑅 by multiplicative system 𝑆

𝑀𝑆

localization of the module 𝑀 by multiplicative system 𝑆

≅

isomorphism

𝑃(𝑀)

the sum of all radical submodules of 𝑀

𝑀

factor module

𝑁

III

Introduction

Introduction Many Authors have studied certain types of modules such as, amply weak supplemented modules, amply FWS modules, amply weak Rad-supplemented modules, On radical supplemented modules, On Rad− ⊕ −supplemented modules and some other types. In (2005) Nebiyev, C. studied amply weak supplemented modules and Bilhan, G. studied amply FWS modules. In (2009) Turkmen, E. and Pancar, A. studied On radical supplemented modules. In (2011) Talebi, Y. and Mahmoudi, A. studied On ad− ⊕ −supplemented modules. In (2012) Akray, I., Jabbar, A., and Sazeedeh, R. studied some conditions under which certain types of modules possess localization property. And In (2012), Jabbar, A. studied locally prime and locally Artinian modules and obtained many properties of each type. The main aim of this work is to study the effect of localization on properties of certain types of modules such as small submodules, semisimple submodules, amply

weak

supplemented

modules,

Rad-supplemented

modules,

𝑤−

supplemented modules and some other types. This study consists of three chapters. The first chapter embraces some basic preliminaries and it consists of three sections. In the first section, the concept of localization of commutative rings and modules at multiplicatively system are investigated. In the second section, some basic definitions and examples are given. In the third section, some known results are given on which we depend to prove the results of the other chapters and most of the remarked definitions are supported by examples whenever they are necessary. The second chapter consists of two sections. In section one, some properties of certain types of modules, that are preserved under localization are established. For example, it is proved that under certain condition, amply cofinitely weak supplementedness property of modules can be transformed from the localization of a given module in to the module itself. Also, some new algebraic structures such as 1

Introduction

amply weak supplemented modules and amply cofinitely weak supplemented modules are studied. In section two, some results concerning localization of modules are proved. It also studies the effect of localization on certain types of modules such as Rad-supplemented modules and weak Rad-supplemented modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization. Chapter three consists of two sections. In the first section, some results concerning localization of modules are proved. It also studies the effect of localization on certain types of modules such as small submodules, lie above submodules and semi simple modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization. In the second section, some results concerning the localization of the modules are proved. It also studies the effect of localization on certain types of modules such as supplement submodules, w-supplemented modules and weak Rad-supplemented modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization.

2

Chapter One The Fundamentals

Chapter One

The Fundamentals

Chapter One The Fundamentals 1.1 The Construction of 𝑅𝑆 and 𝑀𝑆 Let 𝑅 be a commutative ring with identity and 𝑀 be an 𝑅 −module. A nonempty set 𝑆 of 𝑅 is called multiplicative system in 𝑅 if 0 ∉ 𝑆 and 𝑎, 𝑏 ∈ 𝑆 implies that 𝑎𝑏 ∈ 𝑆 [10]. A relation (~) on 𝑅 × 𝑆 is defined as follows: For (𝑎, 𝑠), (𝑏, 𝑡) ∈ 𝑅 × 𝑆, we define (𝑎, 𝑠)~(𝑏, 𝑡) if and only if there exists 𝑣 ∈ 𝑆 such that 𝑣(𝑠𝑏 − 𝑡𝑎) = 0. It can be shown that this relation is an equivalence relation on 𝑅 × 𝑆. The equivalence class of an element (𝑎, 𝑠) ∈ 𝑅 × 𝑆 is denoted by

𝑎 𝑠

, so that

𝑎

= {(𝑏, 𝑡): (𝑏, 𝑡)~(𝑎, 𝑠)} and the set of all such

𝑠

𝑎

equivalence classes is denoted by 𝑆 −1 𝑅 (or 𝑅𝑆 ), so that 𝑅𝑆 = { : (𝑎, 𝑠) ∈ 𝑅 × 𝑠

𝑎

𝑏

𝑡𝑎+𝑠𝑏

𝑠

𝑡

𝑠𝑡

𝑆}. On 𝑅𝑆 , we define: + =

and

𝑎𝑏 𝑠 𝑡

=

𝑎𝑏 𝑠𝑡

, for all

𝑎 𝑏

, ∈ 𝑅𝑆 [10]. It can

𝑠 𝑡 𝑠

be shown that 𝑅𝑆 forms a commutative ring with identity , for 𝑠 ∈ 𝑆 and it is 𝑠

known as " the localization of 𝑅 at the multiplicatively system 𝑆 ". Next, the relation (~) on 𝑀 × 𝑆 is defined: For (𝑎, 𝑠), (𝑏, 𝑡) ∈ 𝑀 × 𝑆, define as (𝑎, 𝑠)~(𝑏, 𝑡) if and only if there exists 𝑣 ∈ 𝑆 such that 𝑣(𝑠𝑏 − 𝑡𝑎) = 0. It can be shown that this relation is an equivalence relation on 𝑀 × 𝑆. The 𝑎

equivalence class of an element (𝑎, 𝑠) ∈ 𝑀 × 𝑆 is denoted by , so that 𝑠

𝑎 𝑠

= {(𝑏, 𝑡): (𝑏, 𝑡)~(𝑎, 𝑠)} and the set of all such equivalence classes is 𝑎

denoted by 𝑆 −1 𝑀 (or 𝑀𝑆 ), so that 𝑀𝑆 = { : (𝑎, 𝑠) ∈ 𝑀 × 𝑆}. On 𝑀𝑆 , define: 𝑠

𝑎 𝑠

𝑏

𝑡𝑎+𝑠𝑏

𝑡

𝑠𝑡

+ =

and

𝑟𝑎 𝑢𝑠

=

𝑟𝑎 𝑢𝑠

, for all

𝑟 𝑢

𝑎 𝑏

∈ 𝑅𝑆 , , ∈ 𝑀𝑆 [10]. It can be shown that 𝑠 𝑡

𝑀𝑆 forms an 𝑅𝑆 −module under these two operations and it is known as the

3

Chapter One

The Fundamentals

localization of 𝑀 at the multiplicatively system 𝑆 of 𝑅. Next, we mention the following remarks. (1) We mention that the ring 𝑅𝑆 has the identity element though 𝑅 does not have. In fact, for any 𝑠 ∈ 𝑆, we have 𝑥 𝑡

𝑠 𝑠

is the identity element of 𝑅𝑆 , since if

∈ 𝑅𝑆 , then as 𝑠((𝑠𝑡)𝑥 − 𝑡(𝑠𝑥)) = 0, so that (𝑠𝑥, 𝑠𝑡)~(𝑥, 𝑡). Hence,

that means

𝑠𝑥 𝑠𝑡

=

𝑠𝑥 𝑠𝑡

𝑠𝑥 𝑠𝑡

𝑥

= , 𝑡

𝑥

= . 𝑡

𝑠

(2) Note that the identity element 𝑠

𝑣

𝑠

𝑣

𝑠

does not depend on the choice of the

elements of 𝑆, that is, = , for all 𝑠, 𝑣 ∈ 𝑆. To prove this, it is known that for all 𝑠, 𝑣 ∈ 𝑆, we have 𝑠(𝑠𝑣 − 𝑣𝑠) = 0, which means that (𝑠, 𝑠)~(𝑣, 𝑣) and thus 𝑠 𝑠

𝑣

= , for all 𝑠, 𝑣 ∈ 𝑆. 𝑣

(3) If 𝑥, 𝑦 ∈ 𝑅 (𝑥, 𝑦 ∈ 𝑀) and 𝑠 ∈ 𝑆, then from (1) we have element of 𝑅𝑆 and so that

𝑥+𝑦 𝑠

=

𝑠 𝑥+𝑦 𝑠

𝑠

=

𝑠(𝑥+𝑦) 𝑠𝑠

=

𝑠𝑥+𝑠𝑦 𝑠𝑠

𝑠 𝑠

𝑥

𝑦

𝑠

𝑠

is the identity

= + .

(4) The result in (4) can be generalized to any 𝑛 elements of 𝑅 (Or 𝑀), that is, if 𝑥1 , 𝑥2 , … , 𝑥𝑛 ∈ 𝑅 (Or 𝑥1 , 𝑥2 , … , 𝑥𝑛 ∈ 𝑀) and 𝑠 ∈ 𝑆, then one can easily get that

𝑥1 +𝑥2 +⋯+𝑥𝑛 𝑠

=

𝑥1 𝑠

+

𝑥2 𝑠

+ ⋯+

𝑥𝑛 𝑠

.

𝑥

(5) If = 0, where 𝑥 ∈ 𝑅 (𝑥 ∈ 𝑀) and 𝑠 ∈ 𝑆, then there exists 𝑡 ∈ 𝑆 such that 𝑠

𝑡𝑥 = 0. To show this, we have

𝑥 𝑠

0

= 0 = , hence (0, 𝑠)~(𝑥, 𝑠), so there exists 𝑠

𝑣 ∈ 𝑆 such that 𝑣(𝑠𝑥 − 𝑠0) = 0, then 𝑣𝑠𝑥 = 0. If we let 𝑡 = 𝑣𝑠, then clearly 𝑡 ∈ 𝑆 and 𝑡𝑥 = 𝑣𝑠𝑥 = 0.

4

Chapter One

The Fundamentals

1.2 Some Definitions and Examples Definition 1.2.1 [1] Let 𝑀 be an 𝑅 −module and 𝑁 ≤ 𝑀. If 𝑟 ∈ 𝑅, then (𝑁: 𝑟) is defined as (𝑁: 𝑟) = {𝑥 ∈ 𝑀 ; 𝑟𝑥 ∈ 𝑁}. Definition 1.2.2 [1] Let 𝑀 be an 𝑅 −module and 𝑁 ≤ 𝑀. Let 𝑆 be a multiplicative system of 𝑅, then (𝑁: 𝑆) is defined as (𝑁: 𝑆) = {𝑥 ∈ 𝑀 ; 𝑆𝑥 ⊆ 𝑁}. Example 1.2.3 Consider the 𝑍 −module 𝑍6 and the submodule 𝑁 = {0̅, 3̅} of 𝑍6 . If we take the multiplicative system 𝑆 = {−1, 1} in 𝑍, then we have (𝑁: −1) = {𝑥 ∈ 𝑍6 ; −1𝑥 ∈ 𝑁} = 𝑁 and also we have (𝑁: 1) = {𝑥 ∈ 𝑍6 ; 1𝑥 ∈ 𝑁} = 𝑁. Hence, (𝑁: 𝑠) = 𝑁 for all 𝑠 ∈ 𝑆. Definition 1.2.4 [6] An 𝑅 −module 𝑀 is called lifting module if every 𝑁 ≤ 𝑀 lies over a direct summand, that is, 𝑁 contains a direct summand 𝐾 ≤ 𝑀 such that 𝑀 𝐾

𝑁 𝐾

≪

. And by [18] 𝑀 is lifting if and only if 𝑀 is amply supplemented and every

supplement submodule of 𝑀 is direct summand of 𝑀. Definition 1.2.5 [18] An 𝑅 −module 𝑀 is called Radical module if it has no maximal submodule, that is, 𝑅𝑎𝑑(𝑀) = 𝑀. And 𝑅𝑎𝑑(𝑀) = intersection of all maximal submodules of 𝑀. Definition 1.2.6 [1] Let 𝑀 be an 𝑅 −module and 𝑃 a prime ideal of , then for 𝐾 ≤ 𝑀 defined 𝑆(𝐾) = {𝑟 ∈ 𝑅; 𝑟𝑚 ∈ 𝐾; for some 𝑚 ∉ 𝑀 − 𝐾}.

5

Chapter One

The Fundamentals

Definition 1.2.7 [1] An 𝑅 −module 𝑀 is called a reduced module. If 𝑃(𝑀) = 0, where 𝑃(𝑀) = the sum of all radical submodules of 𝑀. Definition 1.2.8 [18] Let 𝑀1 , 𝑀2 be submodules of the 𝑅 −module 𝑀. If 𝑀 = 𝑀1 + 𝑀2 and 𝑀1 ∩ 𝑀2 = 0, then 𝑀 is called the (internal) direct sum of 𝑀1 and 𝑀2 . This is written as 𝑀 = 𝑀1 ⨁ 𝑀2 and is called a direct decomposition of 𝑀. In this case every 𝑚 ∈ 𝑀 can be uniquely written as 𝑚 = 𝑚1 + 𝑚2 with 𝑚1 ∈ 𝑀1 , 𝑚2 ∈ 𝑀2 . Moreover 𝑀1 and 𝑀2 are called direct summands of 𝑀. If 𝑀1 is a direct summand, then in general there are various submodules 𝑀2 with 𝑀 = 𝑀1 ⨁ 𝑀2 . 𝑀 is called (direct) indecomposable if 𝑀 ≠ 0 and it cannot be written as a direct sum of non-zero submodules. Observe that 𝑀 = 𝑀1 ⨁ 0 always is a (trivial) decomposition of 𝑀. Definition 1.2.9 [9] A proper submodule 𝑁 of an 𝑅 − module 𝑀 is called a prime submodule of 𝑀 if 𝑟 ∈ 𝑅, 𝑥 ∈ 𝑀such that 𝑟𝑥 ∈ 𝑁, then 𝑥 ∈ 𝑁 or 𝑟𝑀 ⊆ 𝑁 (or, equivalently 𝑥 ∈ 𝑁 or 𝑟 ∈ (𝑁: 𝑀) ). Example 1.2.10 Consider the 𝑍 −module 𝑍15 . It is easy to check that the submodules < 3 > and < 5 > are prime submodules of 𝑍15 . But the submodule < 4 > of the 𝑍 −module 𝑍8 is not prime, since we have 2.2 ∈< 4 > but neither 2 ∈< 4 > nor 2. 𝑍8 ⊆< 4 >.

6

Chapter One

The Fundamentals

Definition 1.2.11 [10] A submodule 𝑉 of an 𝑅 −module 𝑀 is called a small submodule of 𝑀, written by 𝑉 ≪ 𝑀, if 𝐿 ≤ 𝑀 is any submodule such that 𝑉 + 𝐿 = 𝑀, then 𝐿 = 𝑀 (equivalently, if 𝑀 is the only submodule of 𝑀 such that 𝑉 + 𝑀 = 𝑀). Example 1.2.12 Consider the submodule 𝑉 = {0̅, 2̅} of the 𝑍 −module 𝑍4 , since 𝑍4 is the only submodule of 𝑍4 such that 𝑉 + 𝑍4 = {0̅, 2̅} + 𝑍4 = 𝑍4 , then {0̅, 2̅} ≪ 𝑍4 . But the submodule {0̅, 3̅} of the 𝑍 −module 𝑍6 is not a small submodule in 𝑍6 since we have{0̅, 3̅} + {0̅, 2̅, 4̅} = 𝑍6 , but {0̅, 2̅, 4̅} ≠ 𝑍6 . Definition 1.2.13 [3] A submodule 𝑉 of an 𝑅 −module 𝑀 is called a supplement (respectively, a weak supplement) of 𝑈 ≤ 𝑀 if 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑉 (respectively, 𝑈 ∩ 𝑉 ≪ 𝑀). Definition 1.2.14 [5] An 𝑅 −module 𝑀 is called a supplemented (respectively, a weak supplemented) 𝑅 −module if every submodule of 𝑀 has a supplement (respectively, a weak supplement) in 𝑀. Definition 1.2.15 [13] An 𝑅 −module 𝑀 is called an amply supplemented 𝑅 −module if for any 𝑈, 𝑉 ≤ 𝑀 with 𝑀 = 𝑈 + 𝑉, there exists a supplement 𝐾 ≤ 𝑈 such that 𝐾 ≤ 𝑉. Example 1.2.16 ̅̅̅̅} of the Consider the submodules 𝑈 = {0̅, 3̅, 6̅, 9̅} and 𝑉 = {0̅, 2̅, 4̅, 6̅, 8̅, 10 ̅̅̅̅} = 𝑍12 𝑍 −module 𝑍12 , then we have 𝑈 + 𝑉 = {0̅, 3̅, 6̅, 9̅} + {0̅, 2̅, 4̅, 6̅, 8̅, 10 ̅̅̅̅} = {0̅, 6̅} ≪ {0̅, 2̅, 4̅, 6̅, 8̅, 10 ̅̅̅̅} = 𝑉. and 𝑈 ∩ 𝑉 = {0̅, 3̅, 6̅, 9̅} ∩ {0̅, 2̅, 4̅, 6̅, 8̅, 10 Hence, 𝑉 is a supplement submodule of 𝑈 in 𝑍12 .

7

Chapter One

The Fundamentals

Example 1.2.17 Consider the submodule 𝑉 = {0̅, 2̅} of the 𝑍 −module 𝑍4 , then we have 𝑉 + 𝑍4 = {0̅, 2̅} + 𝑍4 = 𝑍4 and 𝑉 ∩ 𝑍4 = {0̅, 2̅} ∩ 𝑍4 = {0̅, 2̅} ≪ 𝑍4 . Hence, 𝑍4 is a weak supplement submodule of {0̅, 2̅} in 𝑍4 . Definition 1.2.18 [5] An 𝑅 −module 𝑀 is called an amply weak supplemented 𝑅 − module if every submodule of 𝑀 has ample supplement in 𝑀. Definition 1.2.19 [10] A submodule 𝑁 of an 𝑅 −module is called finitely generated, if there exists a finite number of elements 𝑥1 , 𝑥2 , … , 𝑥𝑛 which generate 𝑁 and in this case we write 𝑁 =< 𝑥1 , 𝑥2 , … , 𝑥𝑛 > and for each 𝑥 ∈ 𝑁, there exists 𝑟1 , 𝑟2 , … , 𝑟𝑛 ∈ 𝑅 such that 𝑥 = ∑𝑛𝑖=1 𝑟𝑖 𝑥𝑖 . Definition 1.2.20 [12] Let 𝑀 be an 𝑅 −module and 𝐾 ≤ 𝑀. Then 𝐾 is called a cofinite submodule of 𝑀, if

𝑀 𝐾

is finitely generated.

Definition 1.2.21 [12] An 𝑅 −module 𝑀 is called a cofinitely supplemented 𝑅-module, if every cofinite submodule of 𝑀 has a supplement in 𝑀. Definition 1.2.22 [12] An 𝑅 −module 𝑀 is called a cofinitely weak supplemented 𝑅 −module, if every cofinite submodule of 𝑀 has a weak supplement in 𝑀. Definition 1.2.23 [15] A submodule 𝑉 of an 𝑅 −module 𝑀 is called a Rad-supplement or a generalized supplement (respectively, a weak Rad-supplement or a

8

Chapter One

The Fundamentals

generalized weak supplement) of 𝑈 ≤ 𝑀

if 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≤

𝑅𝑎𝑑(𝑉) (respectively, 𝑈 ∩ 𝑉 ≤ 𝑅𝑎𝑑(𝑀)). Definition 1.2.24 [19] An 𝑅 −module 𝑀 is called a Rad-supplemented or a generalized supplemented (respectively, a weakly Rad-supplemented or a generalized weakly supplemented) module if every submodule of 𝑀 has a Radsupplement or a generalized supplement (respectively, a weak Radsupplement or a generalized weak supplement) in 𝑀. Definition 1.2.25 [4] An 𝑅 −module 𝑀 is called completely weak Rad-supplemented, if every submodule of 𝑀 is weakly Rad-supplemented. Definition 1.2.26 [14] An 𝑅 −module 𝑀 is called amply Rad-supplemented (or generalized amply supplemented) in case 𝑀 = 𝑈 + 𝑉 implies that 𝑈 has a Radsupplement (or has a generalized supplement) in 𝑉. Definition 1.2.27 [6] An 𝑅 −module 𝑀 is called a hollow module, if every proper submodule of 𝑀 is small submodule in 𝑀, moreover 𝑀 is called local module, if it is hollow and has a unique maximal submodule, equivalently it is called local module if 𝑅𝑎𝑑(𝑀) is a small maximal submodule of 𝑀. Definition 1.2.28 [6] An 𝑅 −module 𝑀 is called a simple module, if it has no non zero proper submodule. And it is called semisimple module. If every submodule of 𝑀 is a direct summand of simple submodules.

9

Chapter One

The Fundamentals

Definition 1.2.29 [7] An 𝑅 −module 𝑀 is called 𝒘 −supplemented 𝑹 −module, if every semisimple submodule of 𝑀 has a supplement in 𝑀. Definition 1.2.30 [11] A submodule 𝐿 of an 𝑅 −module 𝑀 is said to lie above a submodule 𝑁 of 𝑀 if 𝑁 ≤ 𝐿 and

𝐿 𝑁

≪

𝑀 𝑁

.

1.3 Some Known Results Theorem 1.3.1 (Modular Law for Submodules) [10] Let 𝐾, 𝐿, and 𝑁 be submodules of an 𝑅 −module M . If 𝐾 ≤ 𝑁 , then 𝐾 + (𝐿 ∩ 𝑁) = (𝐾 + 𝐿) ∩ 𝑁. Lemma 1.3.2 [1] Let 𝑀 be an 𝑅 −module and 𝑆 a multiplicative system in 𝑅. If 𝑁 ′ is a submodule of 𝑀𝑆 , then there exists a submodule 𝑁 of 𝑀 such that 𝑁 ′ = 𝑁𝑆 . Furthermore, 𝑁 is a proper submodule of 𝑀 if and only if 𝑁𝑆 is a proper submodule of 𝑀𝑆 . Proof. Since 𝑆 ≠ ∅, let 𝑠 ∈ 𝑆 be any element and 𝑁 = {𝑥 ∈ 𝑀; that

𝑠𝑥 𝑠

=

𝑡𝑥

𝑠𝑥 𝑠

∈ 𝑁 ′ } (note

for all 𝑠, 𝑡 ∈ 𝑆, so there is no any confusion if we take any other

𝑡

elements such 𝑡 ∈ 𝑆 to define 𝑁). We will show that 𝑁 is a submodule of 𝑀. Clearly 0 = have

𝑠(𝑥−𝑦) 𝑠

𝑠0 𝑠

=

∈ 𝑁 ′ , so that ∅ ≠ 𝑁 ⊆ 𝑀. Let 𝑥, 𝑦 ∈ 𝑁 and 𝑟 ∈ 𝑅. Then we

𝑠𝑥−𝑠𝑦 𝑠

=

𝑠𝑥 𝑠

−

𝑠𝑦 𝑠

∈ 𝑁 ′, so that 𝑥 − 𝑦 ∈ 𝑁 and

𝑠(𝑟𝑥) 𝑠

=

𝑠𝑟 𝑠𝑥 𝑠

.

𝑠

∈ 𝑁 ′,

so that 𝑟𝑥 ∈ 𝑁. Thus 𝑁 is a submodule of 𝑀. To show that 𝑁 ′ = 𝑁𝑆 , let 𝑥 ′ ∈ 𝑁 ′ , then 𝑥 ′ =

𝑥 𝑢

for some 𝑥 ∈ 𝑀 and 𝑢 ∈ 𝑆. Now we have

10

𝑠𝑥 𝑠

=

𝑢(𝑠𝑥) 𝑢𝑠

=

Chapter One

𝑢𝑠 𝑥 𝑠

. = 𝑢

𝑢𝑠 𝑠

The Fundamentals

. 𝑥 ′ ∈ 𝑁 ′. Thus 𝑥 ∈ 𝑁, so that 𝑥 ′ =

if 𝑥 ′ ∈ 𝑁𝑆 , then 𝑥 ′ = 𝑥′ =

𝑛 𝑣

1 𝑠𝑛

= . 𝑣

𝑠

𝑛 𝑣

𝑥 𝑢

∈ 𝑁𝑆 and hence 𝑁 ′ ⊆ 𝑁𝑆 and

for some 𝑛 ∈ 𝑁 and 𝑣 ∈ 𝑆, it is clear

𝑠𝑛

∈ 𝑁 ′ , so that

𝑠

∈ 𝑁 ′. Thus 𝑁𝑆 ⊆ 𝑁 ′. Hence, 𝑁 ′ = 𝑁𝑆 . Now let 𝑁 ≠ 𝑀. If

𝑁𝑆 = 𝑀𝑆 , then for any 𝑥 ∈ 𝑀 we have and 𝑙 ∈ 𝑆, thus

𝑠𝑎 𝑠

∈ 𝑁 ′ , then

𝑠𝑥 𝑠

𝑥 𝑠

∈ 𝑁𝑆 , so that

𝑠 𝑠𝑥

= . 𝑠

𝑠

=

𝑥 𝑠

=

𝑎

for some 𝑎 ∈ 𝑁

𝑙

𝑠𝑠 𝑥

𝑠𝑠 𝑎

𝑠 𝑠𝑎

𝑠

𝑠

𝑙

. = 𝑠

. = . 𝑙

𝑠

∈ 𝑁 ′ . Thus

𝑥 ∈ 𝑁, so that 𝑁 = 𝑀 which is a contradiction so 𝑁𝑆 ≠ 𝑀𝑆 . Now, let 𝑁𝑆 ≠ 𝑀𝑆 and if 𝑁 = 𝑀. Then, clearly 𝑁𝑆 = 𝑀𝑆 which is a contradiction. Hence 𝑁 ≠ 𝑀. Proposition 1.3.3 [9] Let 𝐿 and 𝑁 be submodules of an 𝑅−module 𝑀. Then 𝐿 ⊆ 𝑁 if and only if 𝐿𝑃 ⊆ 𝑁𝑃 for every maximal ideal 𝑃 of 𝑅. In particular, we have 𝐿 = 𝑁 if and only if 𝐿 ⊆ 𝑁 and 𝑁 ⊆ 𝐿 if and only if 𝐿𝑃 ⊆ 𝑁𝑃 and 𝑁𝑃 ⊆ 𝐿𝑃 for every maximal ideal 𝑃 of 𝑅 if and only if 𝐿𝑃 = 𝑁𝑃 for every maximal ideal 𝑃 of 𝑅. Proof. Let 𝐿 ⊆ 𝑁 and 𝑃 is any maximal ideal of 𝑅. To show 𝐿𝑃 ⊆ 𝑁𝑃 . If for 𝑥 ∈ 𝑀 and 𝑝 ∉ 𝑃, then

𝑥 𝑝

=

𝑦 𝑞

𝑥 𝑝

∈ 𝐿𝑃 ,

for some 𝑦 ∈ 𝐿 and 𝑞 ∉ 𝑃, this implies that

𝑠𝑞𝑥 = 𝑠𝑝𝑦 ∈ 𝐿, for some 𝑠 ∉ 𝑃. Then we have

𝑥 𝑝

𝑠 𝑞 𝑥

𝑥𝑞𝑥

𝑠 𝑞 𝑝

𝑠𝑞𝑝

= . . =

∈ 𝑁𝑃 , so

that 𝐿𝑃 ⊆ 𝑁𝑃 . Conversely, suppose that the condition of the proposition holds. Let 𝑥 ∈ 𝐿 and 𝑃 be any maximal ideal of 𝑅, then we have 𝑥 1

𝑥

𝑦

1

𝑟

∈ 𝐿𝑃 ⊆ 𝑁𝑃 , that is = , for some 𝑦 ∈ 𝑁 and 𝑟 ∉ 𝑃, which implies that

there exists 𝑠 ∉ 𝑃 such that 𝑠𝑟𝑥 = 𝑠𝑦 ∈ 𝑁, where 𝑠𝑟 ∉ 𝑃. Put 𝑠𝑟 = 𝑟𝑃 , so we have established that for each maximal ideal 𝑃 of 𝑅 there exists 𝑟𝑃 ∉ 𝑃 such that 𝑟𝑝 𝑥 ∈ 𝑁. Let 𝑆 = {𝑟𝑝 ∶ 𝑃 is a maximal ideal of 𝑅} and 𝐴 =< 𝑆 >. If 𝐴 ≠ 𝑅, then there exists a maximal ideal 𝑃0 of 𝑅 such that 𝐴 ⊆ 𝑃0 . In 11

Chapter One

The Fundamentals

this case, we get 𝑟𝑃0 ∈ 𝑃0 , which is a contradiction. Thus 𝐴 = 𝑅, so that 1 ∈ 𝐴 =< 𝑆 >, that means there exists a finite number of maximal ideals 𝑃1 , 𝑃2 , 𝑃3 , … , 𝑃𝑛 of 𝑅 and 𝑟1 , 𝑟2 , 𝑟3 , . . . , 𝑟𝑛 ∈ 𝑅 such that 1 = 𝑟1 𝑟𝑃1 + 𝑟2 𝑟𝑃2 + 𝑟3 𝑟𝑃3 +. . . +𝑟𝑛 𝑟𝑃𝑛 .

Then,

we

have

𝑥 = 1. 𝑥 = 𝑟1 𝑟𝑃1 𝑥 + 𝑟2 𝑟𝑃2 𝑥 +

𝑟3 𝑟𝑃3 𝑥+. . . +𝑟𝑛 𝑟𝑃𝑛 𝑥 ∈ 𝑁. Hence, 𝐿 ⊆ 𝑁. Proposition 1.3.4 [1] Let 𝑀 be an 𝑅 −module, 𝑆 be a multiplicative system in 𝑅 and 𝑁 be a proper submodule of 𝑀. If for each proper submodule 𝐾 of 𝑀, we have (𝐾: 𝑠) = 𝐾 for all 𝑠 ∈ 𝑆, then (1) 𝑁 ≪ 𝑀 if and only if 𝑁𝑆 ≪ 𝑀𝑆 . (2) 𝑁 is a supplemented submodule of 𝑀 if and only if 𝑁𝑆 is a supplemented submodule of 𝑀𝑆 . (3) 𝑀 is a hollow 𝑅 −module if and only if 𝑀𝑆 is a hollow 𝑅𝑆 −module. (4) 𝑀 is a lifting 𝑅 −module if and only if 𝑀𝑆 is a lifting 𝑅𝑆 −module. Proof. (1) Let 𝑁 ≪ 𝑀. Suppose that 𝐿′ is a submodule of 𝑀𝑆 such that 𝑁𝑆 + 𝐿′ = 𝑀𝑆 , then by Lemma 1.3.2, 𝐿′ = 𝐿𝑆 for some submodule 𝐿 of 𝑀 so 𝑁𝑆 + 𝐿𝑆 = 𝑀𝑆 which gives that (𝑁 + 𝐿)𝑆 = 𝑀𝑆 . If 𝑥 ∈ 𝑀, then for 𝑠 ∈ 𝑆 (since 𝑆 ≠ 𝜑) we have

𝑥 𝑠

∈ 𝑀𝑆 and thus

𝑥 𝑠

=

𝑛+𝑙 𝑡

for some 𝑛 ∈ 𝑁, 𝑙 ∈ 𝐿 and 𝑡 ∈ 𝑆, so we get

𝑞𝑡𝑥 = 𝑞𝑠𝑛 + 𝑞𝑠𝑙 ∈ 𝑁 + 𝐿 and thus 𝑥 ∈ (𝑁 + 𝐿): 𝑝 = 𝑁 + 𝐿. Hence, 𝑀 ⊆ 𝑁 + 𝐿. So 𝑁 + 𝐿 = 𝑀. As 𝑁 is small, we get 𝐿 = 𝑀 and thus 𝐿′ = 𝐿𝑆 = 𝑀𝑆 so that 𝑁𝑆 ≪ 𝑀𝑆 . Conversely, let 𝑁𝑆 ≪ 𝑀𝑆 . If 𝐿 is a submodule of 𝑀 with 𝑁 + 𝐿 = 𝑀, then 𝑁𝑆 + 𝐿𝑆 = (𝑁 + 𝐿)𝑆 = 𝑀𝑆 . As 𝑁𝑆 ≪ 𝑀𝑆 we get 𝐿𝑆 = 𝑀𝑆 . Now, if 𝑚 ∈ 𝑀,

12

Chapter One

The Fundamentals

then for any 𝑠 ∈ 𝑆, we have

𝑚 𝑠

∈ 𝐿𝑆 , that means

𝑚 𝑠

=

𝑙 𝑡

for some 𝑙 ∈ 𝐿 and

𝑡 ∈ 𝑆, then for some 𝑞 ∈ 𝑆, we have 𝑞𝑡𝑚 = 𝑞𝑠𝑙, thus 𝑚 ∈ 𝐿: 𝑞𝑡 = 𝐿 so that 𝐿 = 𝑀 and thus 𝑁 ≪ 𝑀. (2) Let 𝑁 be a supplemented submodule of 𝑀. So that, 𝑁 is a supplement of some submodule 𝐿 of 𝑀 and thus 𝑁 + 𝐿 = 𝑀 and 𝑁 ∩ 𝐿 ≪ 𝑁, then 𝑁𝑆 + 𝐿𝑆 = (𝑁 + 𝐿)𝑆 = 𝑀𝑆 and by part (1), we get 𝑁𝑆 ∩ 𝐿𝑆 = (𝑁 ∩ 𝐿)𝑆 ≪ 𝑀𝑆 so that 𝑁𝑆 is a supplement submodule of 𝐿𝑆 in 𝑀𝑆 and thus 𝑁𝑆 is a supplemented submodule of 𝑀𝑆 . (3) Let 𝑀 be a hollow 𝑅 −module and 𝑁 ′ be any proper submodule of 𝑀𝑆 . Then 𝑁 ′ = 𝑁𝑆 for some submodule 𝑁 of 𝑀. As 𝑁𝑆 is proper in 𝑀𝑆 , we get 𝑁 is proper in 𝑀 so that 𝑁 ≪ 𝑀 and thus by part (1), we get 𝑁 ′ = 𝑁𝑆 ≪ 𝑀𝑆 . Hence, 𝑀𝑆 is a hollow 𝑅 −module. (4) Let 𝑀 be a lifting 𝑅 −module. To show that 𝑀𝑆 is a lifting 𝑅 −module, let 𝑁 ′ be any submodule of 𝑀𝑆 . So that, 𝑁 ′ = 𝑁𝑆 for some submodule 𝑁 of 𝑀. As 𝑀 is lifting, there exist submodules 𝐾, 𝐿 of 𝑀 such that 𝑀 = 𝐾 ⊕ 𝐿 with 𝐾 ≤ 𝑁 and 𝑁 ∩ 𝐿 ≪ 𝐿. Clearly, 𝑀 = 𝐾 + 𝐿, so that 𝑀𝑆 = (𝐾 + 𝐿)𝑆 = 𝐾𝑆 + 𝐿𝑆 and if

𝑥 𝑠

∈ 𝐾𝑆 ∩ 𝐿𝑆 where 𝑥 ∈ 𝑀, 𝑠 ∈ 𝑆, then

𝑥 𝑠

=

𝑘 𝑡

=

𝑙 𝑢

for some

𝑘 ∈ 𝐾, 𝑙 ∈ 𝐿 and 𝑡, 𝑢 ∈ 𝑆. Hence, there exist 𝑝, 𝑞 ∈ 𝑆 such that 𝑝𝑡𝑥 = 𝑝𝑠𝑘 ∈ 𝐾 and 𝑞𝑢𝑥 = 𝑞𝑠𝑙 ∈ 𝐿, thus 𝑝𝑞𝑢𝑡𝑥 ∈ 𝐾 and 𝑝𝑞𝑢𝑡𝑥 ∈ 𝐿, so that 𝑝𝑞𝑢𝑡𝑥 ∈ 𝐾 ∩ 𝐿 = {0} and then 𝑝𝑞𝑢𝑡𝑥 = 0,

𝑥 𝑠

=

𝑝𝑞𝑢𝑡𝑥 𝑝𝑞𝑢𝑡𝑠

= 0, so that 𝐾𝑆 ∩ 𝐿𝑆 = {0}. Hence,

𝑀𝑆 = 𝐾𝑆 ⊕ 𝐿𝑆 . Also we have 𝐾𝑆 ≤ 𝑁𝑆 and 𝑁𝑆 ∩ 𝐿𝑆 = (𝑁 + 𝐿)𝑆 ≪ 𝐿𝑆 . Hence, 𝑀𝑆 is a lifting 𝑅 −module. Conversely, suppose that 𝑀𝑆 is a lifting 𝑅𝑆 −module. Let 𝑁 be any submodule of 𝑀. Then 𝑁𝑆 is a submodule of 𝑀𝑆 . Hence, there exist

13

Chapter One

The Fundamentals

submodules 𝐾𝑆 and 𝐿𝑆 of 𝑀𝑆 where 𝐾, 𝐿 are submodules of 𝑀, such that 𝑀𝑆 = 𝐾𝑆 ⊕ 𝐿𝑆 with 𝐾𝑆 ≤ 𝑁𝑆 and 𝑁𝑆 ∩ 𝐿𝑆 ≪ 𝐿𝑆 . We procced as in the first part and we get 𝑀 = 𝐾 ⊕ 𝐿 with 𝐾 ≤ 𝑁 and 𝑁 ∩ 𝐿 ≪ 𝐿. Hence, 𝑀 is a lifting 𝑅 −module. Corollary 1.3.5 [2] Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅. For submodules 𝑁, 𝐿 of 𝑀 (1) (𝑀/𝑁)𝑃 ≅ 𝑀𝑃 /𝑁𝑃 . (2) (𝑁 + 𝐿)𝑃 = 𝑁𝑃 + 𝐿𝑃 . (3) (𝑁 ∩ 𝐿)𝑃 = 𝑁𝑃 ∩ 𝐿𝑃 . Corollary 1.3.6 [1] Let 𝑀 be an 𝑅 −module and 𝑃 which is a prime ideal of 𝑅 such that 𝑆(𝐾) ⊆ 𝑃, for each proper submodule 𝐾 of 𝑀 then: 1) 𝑀 is coatomic if and only if 𝑀𝑃 is coatomic. 2) 𝑀 is reduced 𝑅 −module if and only if 𝑀𝑃 is reduced 𝑅 −module. 3) 𝑀 is a hollow 𝑅 −module if and only if 𝑀𝑃 is a hollow 𝑅 −module. 4) 𝑀 is a lifting 𝑅 −module if and only if 𝑀𝑃 is a lifting 𝑅 −module. 5) 𝑀 is a local 𝑅 −module if and only if 𝑀𝑃 is a local 𝑅 −module. 6) A submodule 𝑁 of 𝑀 is a maximal submodule of 𝑀 if and only if 𝑁𝑃 is a maximal submodule of 𝑀𝑃 . 7) If 𝑁 is a proper submodule of 𝑀, then 𝑆(𝑁) is an ideal of 𝑅 if and only if 𝑆(𝑁𝑃 ) is an ideal of 𝑅𝑃 . 8) If 𝑁 is a proper submodule of 𝑀, then 𝑁 is primal if and only if 𝑁𝑃 is primal.

14

Chapter One

The Fundamentals

9) If 𝑁 is a proper submodule of 𝑀, then 𝑁 is essential in 𝑀 if and only if 𝑁𝑃 is primal in 𝑀𝑃 . If 𝑁 is a proper submodule of 𝑀, then 𝑁 ≪ 𝑀 if and only if

10)

𝑁𝑃 ≪ 𝑀𝑃 . If 𝑁 is a proper submodule of 𝑀, then 𝑁 is a supplemented

11)

submodule of 𝑀 if and only if 𝑁𝑃 is a supplemented submodule of 𝑀𝑃 . If 𝑁 is a proper submodule of 𝑀, then 𝑅𝑎𝑑( 𝑁𝑃 ) = (𝑅𝑎𝑑(𝑁))𝑃 .

12)

Proposition 1.3.7 [8] Let 𝑀 be an 𝑅 −module. If 𝑅 is a local ring with 𝑃 as its unique maximal ideal, then we have 𝑀 ≅ 𝑀𝑃 . Proof. Let

𝑚 𝑝

∈ 𝑀𝑃 , for 𝑚 ∈ 𝑀, 𝑝 ∉ 𝑃, then we get 𝑝 which is a unit in 𝑅, so that 𝑚

𝑝−1 ∈ 𝑅. Now, define 𝑓: 𝑀𝑃 → 𝑀 by 𝑓 ( ) = 𝑝−1 𝑚. One can easily prove 𝑝 that 𝑓 is an isomorphism, so that 𝑀 ≅ 𝑀𝑃 . Lemma 1.3.8 [3] 1)

Let 𝑋 ≪ 𝑀, then any submodule of 𝑋 is also small in 𝑀.

2)

Let 𝑋 ≤ 𝑌 ≤ 𝑀. If 𝑋 ≪ 𝑌, then 𝑋 ≪ 𝑀.

3)

If 𝑓: 𝑀 → 𝑁 is a homomorphism and 𝐿 ≪ 𝑀, then 𝑓(𝐿) ≪ 𝑓(𝑀).

4)

Let 𝑀 be an 𝑅 −module and 𝑋 ≪ 𝑀. Let 𝑋 ≤ 𝐴 ≤ 𝑀 and 𝐴 be a supplement in 𝑀, then 𝑋 ≪ 𝐴 too.

Lemma: 1.3.9 [12] Every supplement submodule of an amply weak supplemented module is amply weak supplemented.

15

Chapter One

The Fundamentals

Proof. Let 𝑀 be an amply weak supplemented module and 𝑉 be any supplement submodule of 𝑀. Let 𝑉 be a supplement of 𝑈 in 𝑀. Let 𝐾 ≤ 𝑉 and 𝐾 + 𝑇 = 𝑉 for 𝑇 ≤ 𝑉. Then, 𝑈 + 𝐾 + 𝑇 = 𝑀. Since 𝑀 is amply weak supplemented module, 𝑈 + 𝐾 has a weak supplement 𝑇 ′ in 𝑀 with 𝑇 ′ ≤ 𝑇. This case 𝑈 + 𝐾 + 𝑇 ′ = 𝑀 and (𝑈 + 𝐾) ∩ 𝑇 ′ ≪ 𝑀.Since 𝐾 + 𝑇 ′ ≤ 𝑉 and 𝑉 is a supplement of 𝑈 in 𝑀, 𝐾 + 𝑇 ′ = 𝑉. To prove that 𝐾 ∩ 𝑇 ′ ≪ 𝑉. Let 𝐾 ∩ 𝑇 ′ + 𝑆 = 𝑉 for 𝑆 ≤ 𝑉. Since 𝑈 + 𝐾 ∩ 𝑇 ′ + 𝑆 = 𝑀 and 𝐾 ∩ 𝑇 ′ ≤ (𝑈 + 𝐾) ∩ 𝑇 ′ ≪ 𝑀, 𝑈 + 𝑆 = 𝑀. Since 𝑆 ≤ 𝑉 and 𝑉 is a supplement of 𝑈 in 𝑀, 𝑆 = 𝑉. Hence 𝐾 ∩ 𝑇 ′ ≪ 𝑉 and 𝑇 ′ is a weak supplement of 𝐾 in 𝑉. Since 𝑇 ′ ≤ 𝑇, 𝐾 has ample weak supplements in 𝑉. Thus, 𝑉 is amply weak supplemented. Lemma 1.3.10 [3] Let 𝑀 be an 𝑅 −module with 𝑈, 𝑉 ≤ 𝑀, and let 𝑉 be a weak supplement submodule of 𝑈 in 𝑀. Then, for 𝐿 ≤ 𝑈,

𝑉+𝐿 𝐿

is a weak supplement of

𝑈 𝐿

𝑀

in . 𝐿

Proof. Since 𝑉 is a weak supplement submodule of 𝑈 in 𝑀, then 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑀. Then

𝑀

𝑈

=

𝐿

∩

𝑉+𝐿 𝐿

𝑈∩𝑉+𝐿 𝐿

=

𝑈∩(𝑉+𝐿) 𝐿

𝐿

=

𝑈+𝑉 𝐿

𝑈∩𝑉+𝐿 𝐿

=

𝑈 𝐿

+

𝑉+𝐿 𝐿

. Now with the help of the modular law

. Since 𝑈 ∩ 𝑉 ≪ 𝑀, then by Lemma 1.3.8,

𝑀

≪ . 𝐿

Lemma 1.3.11 [12] Every factor module of an amply cofinitely weak supplemented module is amply cofinitely weak supplemented.

16

Chapter One

The Fundamentals

Proof. Let 𝑀 be an amply cofinitely weak supplemented module and factor module of 𝑀. Let generated. By

𝑀 𝐾

𝑀 𝑁 𝐾 𝑁

≅

submodule of 𝑀. For

𝑉 𝑁

,

𝐾 𝑁

𝑀 𝐾

≤

be a cofinite submodule of

𝑀 𝑁

. Then,

𝑀 𝑁 𝐾 𝑁

𝑀 𝑁

be any

is finitely

is also finitely generated. Hence, 𝐾 is a cofinite 𝑀 𝑁

, let

𝐾 𝑁

+

𝑉 𝑁

𝑀

= . Then, 𝐾 + 𝑉 = 𝑀 and because 𝑀 𝑁

is an amply cofinitely weak supplemented module and 𝐾 is a cofinite submodule of 𝑀, there exists a weak supplement 𝑉 ′ of 𝐾 with 𝑉 ′ ≤ 𝑉. According to Lemma 1.3.10, this case 𝑉 ′ +𝑁

Since

𝑁

𝑉 ′ +𝑁 𝑁

is a weak supplement of

𝑉 𝐾

𝑀

𝑁𝑁

𝑁

≤ , has ample weak supplements in

. Thus,

𝑀 𝑁

𝐾 𝑁

𝑀

in . 𝑁

is an amply

cofinitely weak supplemented module. Lemma 1.312 [12] Every factor module of an amply weak supplemented module is amply weak supplemented. Proof. Let 𝑀 be an amply weak supplemented module and module of 𝑀. Let

𝐾 𝑁

𝑀

𝑉

𝑁

𝑁

≤ . For

𝑀

𝐾

𝑁

𝑁

≤ , let

+

𝑉 𝑁

𝑀 𝑁

be any factor

𝑀

= . Then 𝐾 + 𝑉 = 𝑀 and 𝑁

because 𝑀 is an amply weak supplemented module, there exists a weak supplement 𝑉 ′ of 𝐾 with 𝑉 ′ ≤ 𝑉. This case by Lemma 1.3.10, supplement of Thus,

𝑀 𝑁

𝐾 𝑁

𝑀

𝑉 ′ +𝑁

𝑁

𝑁

in . Since

𝑉 𝐾

≤ ,

𝑁 𝑁

𝑁

is a weak 𝑀

has ample weak supplements in .

is an amply weak supplemented module.

17

𝑉 ′ +𝑁

𝑁

Chapter One

The Fundamentals

Lemma 1.3.13 [11] 1) Let 𝐿, 𝑁 ≤ 𝑀 and 𝑁 ≤ 𝐿. Then, 𝐿 lies above 𝑁 if and only if 𝑁 + 𝑇 = 𝑀 for every 𝑇 ≤ 𝑀, such that 𝐿 + 𝑇 = 𝑀. 2) A submodule 𝑉 of an 𝑅 −module 𝑀 is a supplement of 𝑈 ≤ 𝑀 if and only if 𝑈 + 𝑉 = 𝑀 and 𝑈 ∩ 𝑉 ≪ 𝑉. Corollary 1.3.14 [12] 1) Every homomorphic image of an amply weak supplemented module is amply weak supplemented module. 2) Every direct summand of an amply weak supplemented module is amply weak supplemented module. 3) Every homomorphic image of an amply cofinitely weak supplemented module is amply cofinitely weak supplemented module. 4) Every direct summand of an amply cofinitely weak supplemented module is amply cofinitely weak supplemented module. Proposition 1.3.15 [19] 1) Let 𝑀 be a Rad-supplement 𝑅 −module and 𝐿 ≤ 𝑀 with 𝐿 ∩ 𝑅𝑎𝑑(𝑀) = 0. Then, 𝐿 is semisimple. In particular, a Radsupplement 𝑅 −module 𝑀 with 𝑅𝑎𝑑(𝑀) = 0 is semisimple. 2) Let 𝑈 and 𝑉 be Rad-supplement 𝑅 −modules. If 𝑀 = 𝑈 + 𝑉 then, 𝑀 is a Rad-supplement 𝑅 −module. Proposition 1.3.16 [19] Let 𝑀 be a weak Rad-supplemented𝑅 −module. Then (1) Every supplement submodule of 𝑀 is a weak Rad-supplemented 𝑅 −module.

18

Chapter One

The Fundamentals

(2) If 𝑓: 𝑁−→ 𝑀 is a cover of 𝑀, 𝑁 is also a weak Rad-supplemented 𝑅 −module. (3) Every factor module of 𝑀 is a weak Rad-supplemented 𝑅 −module. Lemma 1.3.17 [7] Let 𝑀 = 𝑁 + 𝐿 where, 𝐿 ≤ 𝑀 and 𝑁 is a semisimple submodule of 𝑀, then, 𝑀 = 𝑁 ′ ⊕ 𝐿 for some 𝑁 ′ ≤ 𝑁. Proposition 1.3.18 [7] Let 𝑀 be an 𝑅 −module. Then the following statements are equivalent. 1. 𝑀 is w-supplemented 𝑅 −module. 2. Every semisimple submodule of 𝑀 has a supplement that is a direct summand. 3. Every semisimple submodule of 𝑀 has a weak supplement. 4. Every semisimple submodule of 𝑀 has a Rad-supplement. Lemma 1.3.19 [16] Let 𝑀 be an 𝑅 −module and 𝑉 be a Rad-supplement submodule of 𝑈 in 𝑀. If 𝑈 ∩ 𝑉 is a supplement submodule in 𝑈, then 𝑉 is supplement submodule in 𝑀. Proof. By the hypothesis, 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≤ 𝑅𝑎𝑑(𝑉), let 𝑈 ∩ 𝑉 be a supplement of some submodule 𝑋 in 𝑈. Then, 𝑈 = 𝑋 + 𝑈 ∩ 𝑉 and 𝑋 ∩ (𝑈 ∩ 𝑉) = 𝑋 ∩ 𝑉 ≪ 𝑈 ∩ 𝑉. Note that 𝑀 = 𝑈 + 𝑉 = 𝑋 + 𝑈 ∩ 𝑉 + 𝑉 = 𝑋 + 𝑉 and it follows from [6], that 𝑋 ∩ 𝑉 ≪ 𝑉. Hence, 𝑉 is a supplement of 𝑋 in 𝑀. Lemma 1.3.20 [11] Let 𝑉 be a supplement submodule of 𝑈 in 𝑀, 𝐾, 𝑇 ≤ 𝑉. Then, 𝑇 is a supplement of 𝐾 in 𝑉 if and only if 𝑇 is a supplement of 𝑈 + 𝐾 in 𝑀.

19

Chapter One

The Fundamentals

Proof. Let 𝑇 be a supplement of 𝐾 in 𝑉. Let 𝑈 + 𝐾 + 𝐿 = 𝑀 for 𝐿 ≤ 𝑇. Then, 𝐾 + 𝐿 ≤ 𝑉 and because 𝑉 is a supplement of 𝑈, 𝐾 + 𝐿 = 𝑉. Since 𝐿 ≤ 𝑇 and 𝑇 is a supplement of 𝐾 in 𝑉, 𝐿 = 𝑇. Hence, 𝑇 is a supplement of 𝑈 + 𝐾 in 𝑀. (←) Let 𝑇 be a supplement of 𝑈 + 𝐾 in 𝑀. Then by Lemma 1.3.13 𝑈 + 𝐾 + 𝑇 = 𝑀 and (𝑈 + 𝐾) ∩ 𝑇 ≪ 𝑇. Since 𝑈 + 𝐾 + 𝑇 = 𝑀 and 𝐾 + 𝑇 ≤ 𝑉, then we have 𝐾 + 𝑇 = 𝑉. Since 𝐾 ∩ 𝑇 ≤ (𝑈 + 𝐾) ∩ 𝑇 ≪ 𝑇, 𝐾 ∩ 𝑇 ≪ 𝑇. Then, by Lemma 1.3.13, 𝑇 is a supplement of 𝐾 in 𝑉. Lemma 1.3.21 [19] Let 𝑀 be a 𝑅 −module and 𝐾 a supplement submodule of 𝑀. Then 𝐾 ∩ 𝑅𝑎𝑑(𝑀) = 𝑅𝑎𝑑(𝐾). Lemma 1.3.22 [4] Let 𝑀 be an 𝑅 −module, and 𝑉 a weak Rad-supplement of 𝑈 in 𝑀. Then, 𝑉+𝐿 𝐿

is a weak Rad-supplement of

𝑈 𝐿

𝑀

in , for every 𝐿 ≤ 𝑈. 𝐿

Theorem 1.3.23 [11] Let 𝑈 ≤ 𝑀, 𝐿 ≤ 𝑈 and 𝑈 lies above 𝐿. If 𝑈 and 𝐿 have supplements in 𝑀, Then, they have the same supplements in 𝑀. Proof. Let 𝑉 be a supplement submodule of 𝑈 in 𝑀. Then 𝑈 + 𝑉 = 𝑀 and by Lemma 1.3.13, 𝐿 + 𝑉 = 𝑀. Since 𝑉 is a supplement submodule of 𝑈 and 𝐿 ≤ 𝑈, 𝐿 ∩ 𝑉 ≤ 𝑈 ∩ 𝑉 ≪ 𝑉. Thus, 𝑉 is a supplement of 𝐿. Let 𝑇 be a supplement of 𝐿 in 𝑀. Then, 𝐿 + 𝑇 = 𝑀 and by 𝐿 ≤ 𝑈, 𝑈 + 𝑇 = 𝑀. Let 𝑈 + 𝑆 = 𝑀 for some 𝑆 ≤ 𝑇. Then, by Lemma 1.3.13, 𝐿 + 𝑆 = 𝑀 and since 𝑇 is a supplement submodule of 𝐿 in 𝑀, 𝑆 = 𝑇. Thus 𝑇 is a supplement submodule of 𝑈 in 𝑀. 20

Chapter One

The Fundamentals

Theorem 1.3.24 [11] Let 𝑈 ≤ 𝑀, 𝐿 ≤ 𝑈 and 𝑈 lies above 𝐿. If 𝑈 and 𝐿 have weak supplements in 𝑀, then they have the same weak supplements in 𝑀. Theorem 1.3.25 [12] Let 𝑀 be an 𝑅 −module, 𝑈 ≤ 𝑀 and 𝑉 ≤ 𝑀 with 𝑀 = 𝑈 + 𝑉. If 𝑈 and 𝑉 have ample weak supplements in 𝑀, then 𝑈 ∩ 𝑉 also has ample weak supplements in 𝑀. Proof. Let 𝑈 ∩ 𝑉 + 𝑇 = 𝑀 for 𝑇 ≤ 𝑀. Since 𝑈 + 𝑉 = 𝑀, then we can prove that 𝑈 + 𝑉 ∩ 𝑇 = 𝑉 + 𝑈 ∩ 𝑇 = 𝑀. Then by hypothesis 𝑈 has a weak supplement 𝑉 ′ with 𝑉 ′ ≤ 𝑉 ∩ 𝑇 and 𝑉 has a weak supplement 𝑈 ′ with 𝑈 ′ ≤ 𝑈 ∩ 𝑇. This case 𝑈 + 𝑉 ′ = 𝑀, 𝑈 ∩ 𝑉 ′ ≪ 𝑀, 𝑉 + 𝑉 ′ = 𝑀 and 𝑉 ∩ 𝑈 ′ ≪ 𝑀. Since 𝑈 + 𝑉 ′ = 𝑀 and 𝑉 ′ ≤ 𝑉, by Modular law 𝑈 ∩ 𝑉 + 𝑉 ′ = 𝑉. Since 𝑉 + 𝑈 ′ = 𝑀 and 𝑈 ′ ≤ 𝑈, also by Modular law 𝑈 ∩ 𝑉 + 𝑈 ′ = 𝑈. This case 𝑀 = 𝑈 + 𝑉 = 𝑈 ∩ 𝑉 + 𝑈 ′ + 𝑈 ∩ 𝑉 + 𝑉 ′ = 𝑈 ∩ 𝑉 + 𝑈 ′ + 𝑉 ′ . Since (𝑈 ∩ 𝑉) ∩ (𝑈 ′ + 𝑉 ′ ) ≤ (𝑈 ∩ 𝑉 + 𝑈 ′ ) ∩ 𝑉 ′ + (𝑈 ∩ 𝑉 + 𝑉 ′ ) ∩ 𝑈 ′ = 𝑈 ∩ 𝑉 ′ + 𝑉 ∩ 𝑈 ′ ≪ 𝑀, 𝑈 ′ + 𝑉 ′ is a weak supplement of 𝑈 ∩ 𝑉 in 𝑀. Since 𝑈 ′ ≤ 𝑇 and 𝑉 ′ ≤ 𝑇, 𝑈 ′ + 𝑉 ′ ≤ 𝑇. Thus, 𝑈 ∩ 𝑉 has ample weak supplements in 𝑀. Proposition 1.3.26 [16] 1) Let 𝑀 be reduced 𝑅 −module. If 𝑀 is Rad-supplemented then, 𝑅𝑎𝑑(𝑀) ≪ 𝑀. 2) Let M be a reduced 𝑅 −module. Suppose every Rad- supplement submodule of 𝑀 is Rad-supplemented. Then, 𝑀 is supplemented. 3) Let 𝑀 be a radical 𝑅 −module. Then, 𝑀 is Rad-supplemented.

21

Chapter One

The Fundamentals

Corollary 1.3.26 [11] Let 𝑈 and 𝑉 be mutual supplements in 𝑀, 𝐿 be a supplement of 𝑆 in 𝑈 and 𝑇 be a supplement of 𝐾 in 𝑉. Then 𝐿 + 𝑇 is a supplement of 𝐾 + 𝑆 in 𝑀. Lemma 1.3.27 [11] Let 𝑉 be a supplement submodule of 𝑈 in 𝑀 and 𝐾, 𝑇 ≤ 𝑉.Then 𝑇 is a Supplement submodule of 𝐾 in 𝑉 if and only if 𝑇 is a supplement submodule of 𝑈 + 𝐾 in 𝑀. Proposition 1.3.28 [17] Let 𝑅 be a noetherian ring and 𝑀 a simply radical 𝑅 −module. If 𝑀 is amply Rad-supplemented, then 𝑀 is hollow radical. In particular, every Radsupplemented proper submodule of 𝑀 is supplemented. Lemma1.3.29 [17] Let 𝑀 be an 𝑅 −module with 𝑈 + 𝑉 = 𝑀, for 𝑈, 𝑉 ≤ 𝑀. If 𝑉 contains a Rad-supplement submodule of 𝑈 in 𝑀, then 𝑈 ∩ 𝑉 has a Rad-supplement submodule in 𝑉. Proposition 1.3.30 [4] Every factor module of a completely weak Rad-supplemented module is completely weak Rad-supplemented. Lemma 1.3.31[1] Let 𝑀 be an 𝑅 −module and 𝑆 is a multiplicative closed set in 𝑅 such that 𝑆(𝐾) ∩ 𝑆 = ∅, for every 𝐾 ≤ 𝑀 , then (1) If 𝑁 is any submodule of 𝑀 , then 𝑑(𝑁𝑆 ) = (𝑅𝑎𝑑𝑁)𝑆 . (2) If 𝑁 and 𝐿 are submodules of , then 𝑁 = 𝐿 if and only if 𝑁𝑆 = 𝐿𝑆 . (3) 𝑃(𝑀𝑆 ) = (𝑃(𝑀))𝑆 .

22

Chapter One

The Fundamentals

Proposition 1.3.32 [19] Let 𝑀 be finitely generated 𝑅 −module. Then 𝑀 is a weak Radsupplemented 𝑅 −module if and only if 𝑀 is a weakly supplemented 𝑅 −module. Proposition 1.3.33 [19] Let 𝑀 be an 𝑅 − module and 𝑀 = 𝑈 + 𝑉. If 𝑈, 𝑉 have generalized ample supplements in 𝑀, then 𝑈 ∩ 𝑉 also has generalized ample supplements in 𝑀. Proposition 1.3.34 [19] Let 𝑁, 𝑈 ≤ 𝑀 and 𝑁 be a generalized supplemented module. If 𝑁 + 𝑈 has a generalized supplement in 𝑀, then so does 𝑈.

23

Chapter Two Amply Weak Supplemented Modules and Rad- Supplemented Modules

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Chapter Two Amply Weak Supplemented Modules and Rad- Supplemented Modules

This chapter consists of two sections. In the first section some results concerning the localization of the modules are proved. It also studies the effect of localization on certain types of modules such as amply weak supplemented modules and amply cofinitely weak supplemented modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization. In the second section, some results concerning the localization of some further types of modules are proved such as Rad-supplemented modules and weak Rad-supplemented modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization.

2.1 Amply Weak Supplemented Modules This section is devoted to the study of amply weak supplemented modules and amply cofinitely weak supplemented modules. The effect of localization on these types of modules are studied and investigated. Also, some conditions are obtained under which certain properties of such modules are preserved under localization. We begin with the following propositions, in which a condition is given that makes the localization of an amply weak supplemented module an amply weak supplemented module.

24

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proposition 2.1.1 Let 𝑀 be an amply weak supplemented 𝑅 −module and 𝑃 be a prime ideal of 𝑅. If for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃, then 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module. Proof. Let 𝑈 ′ be any submodule of 𝑀𝑃 . To show that 𝑈 ′ has ample weak supplements in 𝑀𝑃 . Now, let 𝑉 ′ be a submodule of 𝑀𝑃 such that 𝑈 ′ + 𝑉 ′ = 𝑀𝑃 , then by Lemma 1.3.2

there exist submodules 𝑈 and 𝑉 of 𝑀 such

that 𝑈 ′ = 𝑈𝑃 and 𝑉 ′ = 𝑉𝑃 , and by Corollary 1.3.5, 𝑈 ′ + 𝑉 ′ = 𝑈𝑃 + 𝑉𝑃 = (𝑈 + 𝑉)𝑃 = 𝑀𝑃 , then by using the given condition, we have (𝑈 + 𝑉): 𝑝 = 𝑈 + 𝑉, from which we get 𝑈 + 𝑉 = 𝑀. Now, since 𝑀 is amply weak supplemented 𝑅 −module, so that every submodule of 𝑀 has ample weak supplemented submodule in 𝑀 and thus 𝑈 has ample weak supplements in 𝑀 and as 𝑈 + 𝑉 = 𝑀, then there exists a weak supplement 𝐾 of 𝑈 such that 𝐾 ≤ 𝑉. Then, 𝑈⋂𝐾 ≪ 𝑀. Next, we get 𝐾 ′ ≤ 𝑉 ′ and, by Proposition 1.3.4, we get 𝑈 ′ ⋂𝐾 ′ = 𝑈𝑃 ⋂𝐾𝑃 = (𝑈⋂𝐾)𝑃 ≪ 𝑀𝑃 . That means 𝐾 ′ is a weak supplement of 𝑈 ′ in 𝑀𝑃 , so that 𝑈 ′ has ample weak supplements in 𝑀𝑃 . Hence, 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 − module. Next, we prove that under a certain condition the amply weak supplementedness property of modules can be transformed from the localization of a given module in to the module itself.

25

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proposition 2.1.2 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module, then 𝑀 is an amply weak supplemented 𝑅 −module. Proof. Let 𝑈 ≤ 𝑀. To show that 𝑈 has an ample weak supplement in 𝑀, let 𝑉 ≤ 𝑀 such that 𝑈 + 𝑉 = 𝑀, then 𝑈𝑃 + 𝑉𝑃 = (𝑈 + 𝑉)𝑃 = 𝑀𝑃 . Since, 𝑈𝑃 , 𝑉𝑃 ≤ 𝑀𝑃 and 𝑀𝑃 is an amply weak supplemented 𝑅 −module, so that 𝑈𝑃 has ample weak supplements in 𝑀𝑃 and as 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 , we get that 𝑈𝑃 has ̅ in 𝑀𝑃 with 𝐾 ̅ ≤ 𝑉𝑃 . But then, 𝐾 ̅ = 𝐾𝑃 , a weak supplement, which we call 𝐾 for some submodule 𝐾 ≤ 𝑀 and so that 𝐾𝑃 ≤ 𝑉𝑃 , from which, one can easily get that 𝐾 ≤ 𝑉 so that 𝑈𝑃 + 𝐾𝑃 = 𝑀𝑃 and 𝑈𝑃 ⋂𝐾𝑃 ≪ 𝑀𝑃 , that is, (𝑈⋂𝐾)𝑃 ≪ 𝑀𝑃 , so by Proposition 1.3.4, we get

𝑈⋂𝐾 ≪ 𝑀, so that 𝐾 is a weak

supplement of 𝑈 in 𝑀, that is, 𝑈 has ample weak supplements in 𝑀. Hence, 𝑀 is an amply weak supplemented 𝑅 −module. Now, by combining the above two propositions, we get the following theorem. Theorem 2.1.3 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. Then, 𝑀 is an amply weak supplemented 𝑅 −module if and only if 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module. 26

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. The proof follows directly from Proposition 2.1.1 and Proposition 2.1.2. As a Corollary and Proposition 1.3.7, we have the following result. Corollary 2.1.4 Let 𝑀 be an 𝑅 −module, where 𝑅 is a local ring with 𝑃 as its unique maximal ideal. Then 𝑀 is an amply weak supplemented 𝑅 −module if and only if 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module. Proof. As, 𝑅 is a local ring with 𝑃 as its unique maximal ideal, by Proposition 1.3.7, we have 𝑀 ≅ 𝑀𝑃 , so that the result directly follows. It is known that every factor module of an amply weak supplemented module is an amply weak supplemented module by Lemma 1.3.12. What will happen under a certain condition if we droop the ample weak supplementedness property from the module and give it to the localization of the module? Does the result remains true? The following proposition will answer this question. Proposition 2.1.5 Let 𝑀 be an 𝑅 −module and 𝑁 a submodule of 𝑀. Let 𝑃 be a prime ideal of 𝑅 such that for each submodule 𝐾 of 𝑀 with 𝑁 ⊆ 𝐾, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module, then amply weak supplemented 𝑅 −module.

27

𝑀 𝑁

is an

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. 𝐾

Let 𝑀 𝑁

𝑁

𝑀

𝐾

𝑁

𝑁

≤ . To show

, where

𝑉 𝑁

𝑀

𝐾𝑃

𝑁

𝑁𝑃

≤ . Then,

+

𝑉𝑃 𝑁𝑃

𝐾

𝑉

𝑀

𝑀𝑃

𝑁

𝑁

𝑁

𝑁𝑃

= ( + )𝑃 = ( )𝑃 =

amply weak supplemented module, so 𝑀𝑃 𝑁𝑃

𝑀

𝐾𝑃

𝑁

𝑁𝑃

= ( )𝑃 , this means that,

so that

𝐾𝑃 𝑁𝑃

𝑀

𝑀𝑃

𝑁

𝑁𝑃

( )𝑃 =

+

𝐿𝑃 𝑁𝑃

=

𝑀𝑃 𝑁𝑃

𝑀

𝐾

𝑁

𝑁

has an ample weak supplement in . Let

and

⋂

𝑁𝑃

𝐿

𝑀

𝑁

𝑁

𝑁

𝑁𝑃

𝐿𝑃 𝑁𝑃

≪

𝑀𝑃 𝑁𝑃

. Since, 𝑀𝑃 is an

𝐿𝑃 𝑁𝑃

in

𝑀𝑃 𝑁𝑃

, where 𝐿 ≤ 𝑀, 𝐾

𝐿

𝑁

𝑁

, from which we get ( + )𝑃 =

𝑀 𝑁

and the submodules of 𝑀 which contain 𝑁, so

from 𝐾: 𝑝 = 𝐾, 𝑁 ⊆ 𝐾, one can easily show that for every submodule where 𝑁 ⊆ 𝐾, we have

𝐾 𝑁

𝐾

𝑀

𝐾

𝑁

𝑁

in , so that

𝐾

𝐾

𝑁

𝑁

: 𝑝 = , for all 𝑝 ∉ 𝑃. Hence, we get

by Proposition 1.3.4, we get 𝑁

=

and ( ⋂ )𝑃 ≪ ( )𝑃 . Since, there is a one to one correspondence

between the submodules of

of

𝑁

has an ample weak supplement in

has a weak supplement

𝐾𝑃

𝐾

𝐾𝑃

𝑉

+

𝐾 𝑁

𝐿

𝑀

𝐿

𝑁

𝑁

𝑁

⋂ ≪ . This means,

+

𝐿 𝑁

𝐾

𝑀

𝑁

=

of ,

𝑀 𝑁

𝑁

and

is a weak supplement

𝑀

𝑀

𝑁

𝑁

has ample weak supplements in . Hence,

is an amply

weak supplemented 𝑅 −module. In Corollary 1.3.14, it is proved that a homomorphic image of an amply weak supplemented module is an amply weak supplemented. What will happen, under a certain condition if we droop the amply weak supplementedness property from the module and give it to the localization of the module? that is, does the result remain true? The following Corollary will give the answer to this question.

28

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Corollary 2.1.6 Let 𝑀 and 𝑀′ be 𝑅 −modules and 𝑃 be a prime ideal of 𝑅, such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑓: 𝑀 → 𝑀′ is homomorphism and 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module, then 𝑓(𝑀) is an amply weak supplemented 𝑅 −module. Proof. By Theorem 2.1.3, we get 𝑀 is an amply weak supplemented module and then by Corollary 1.3.14, we get that 𝑓(𝑀) is an amply weak supplemented 𝑅 −module. Next, we prove that under some condition the supplemented submodules became amply weak supplemented, where the localization of the module is amply weak supplemented. Corollary 2.1.7 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module. If for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾, for all 𝑝 ∉ 𝑃, then every supplemented submodule of 𝑀 is amply weak supplemented. Proof. Since 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module, by Theorem 2.1.3, we get 𝑀 which is an amply weak supplemented module and if 𝑈 is any supplemented submodule of 𝑀, then by Lemma: 1.3.9 , we get 𝑈 which is amply weak supplemented.

29

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Now, we prove that under certain condition the directed summand of a module that its localization amply weak supplemented is an amply weak supplemented modules. Corollary 2.1.8 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑀𝑃 is an amply weak supplemented 𝑅𝑃 −module, then every direct summand of 𝑀 is an amply weak supplemented submodule. Proof. By Theorem 2.1.3, we have 𝑀 which is an amply weak supplemented module and if 𝑈 is any direct summand of 𝑀, then, by Corollary 1.3.14, we get 𝑈 is an amply weak supplemented. Lemma 2.1.9 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑈 ≤ 𝑀 such that 𝑈𝑃 has ample weak supplements in 𝑀𝑃 , then 𝑈 has ample weak supplements in 𝑀. Proof. Let 𝑉 be a submodule of 𝑀 be such that 𝑈 + 𝑉 = 𝑀, then 𝑉𝑃 ≤ 𝑀𝑃 and 𝑈𝑃 + 𝑉𝑃 = (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and as 𝑈𝑃 has ample weak supplements in 𝑀𝑃 , ̅ of 𝑈𝑃 in 𝑀𝑃 , where 𝐾 ̅ ≤ 𝑉𝑃 . Also, by there exists a weak supplement 𝐾 ̅ = 𝐾𝑃 , for some 𝐾 ≤ 𝑀, so that 𝑈𝑃 + 𝐾𝑃 = 𝑀𝑃 and Lemma 1.3.2, we have 𝐾 30

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

𝑈𝑃 ⋂𝐾𝑃 ≪ 𝑀𝑃 , which gives (𝑈 + 𝐾)𝑃 = 𝑈𝑃 + 𝐾𝑃 = 𝑀𝑃 and (𝑈⋂𝐾)𝑃 = 𝑈𝑃 ⋂𝐾𝑃 ≪ 𝑀𝑃 . Then, by the given condition we get 𝑈 + 𝐾 = 𝑀 and by ̅ ≤ 𝑉𝑃 , we can easily Proposition 1.3.4, we get 𝑈⋂𝐾 ≪ 𝑀. Now, as 𝐾𝑃 = 𝐾 get that 𝐾 ≤ 𝑉, so that 𝐾 is a weak supplement of 𝑈 in 𝑀. Hence, 𝑈 has ample weak supplements in 𝑀. Next, By using the above lemma, we can generalize the result that proved in Theorem 1.3.25, to the localized modules. Corollary 2.1.10 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. Let 𝑈, 𝑉 ≤ 𝑀 with 𝑀 = 𝑈 + 𝑉. If 𝑈𝑃 and 𝑉𝑃 have ample weak supplements in 𝑀𝑃 , then 𝑈 ∩ 𝑉 also has ample weak supplements in 𝑀. Proof. By Lemma 2.1.9, we get that 𝑈 and 𝑉 which have ample weak supplements in 𝑀 and then by Theorem 1.3.25, we get (𝑈 ∩ 𝑉) which has ample weak supplements in 𝑀. Lemma 2.1.11 Let 𝑀 be an 𝑅 −module with submodules 𝑈 and 𝑉of 𝑀 and let 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑉𝑃 is a weak supplement of 𝑈𝑃 in 𝑀𝑃 , then for submodule 𝐿 of 𝑈 ,

𝑉+𝐿 𝐿

is a weak supplement of

31

𝑈 𝐿

𝑀

in . 𝐿

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. Since 𝑉𝑃 is a weak supplement of 𝑈𝑃 in 𝑀𝑃 , then 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≪ 𝑀𝑃 , which implies that (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑀𝑃 , then we get 𝑈 + 𝑉 = 𝑀 and by Proposition 1.3.4, we get 𝑈 ∩ 𝑉 ≪ 𝑀. Hence, 𝑉 is a weak supplement of 𝑈 in 𝑀. Now since 𝐿 ≤ 𝑈, thus by Lemma 1.3.10, we get

𝑉+𝐿 𝐿

which is a weak supplement of

𝑈 𝐿

𝑀

in . 𝐿

From Lemma 1.3.11, every factor module of an amply cofinitely weak supplemented module is an amply cofinitely weak supplemented module. Now, the questions what will happen if we droop the amply cofinitely weak supplementedness property from the module and give it to its localization? Does the result remain true? The following lemma will give a partial answer this question. Lemma 2.1.12 Let 𝑀 be an 𝑅 −module and 𝑁 be a submodule of 𝑀. Let 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑀𝑃 is an amply cofinitely weak supplemented 𝑅𝑃 −module, then

𝑀 𝑁

is an amply cofinitely weak supplemented module.

Proof. Let 𝐾𝑃 𝑁𝑃

+

𝐾 𝑁

𝑉𝑃 𝑁𝑃

be a cofinite submodule of =

𝑀𝑃 𝑁𝑃

𝑀 𝑁

and for

𝑉 𝑁

𝑀

𝐾

𝑉

𝑁

𝑁

𝑁

≤ , let +

𝑀

= . Then, 𝑁

such that 𝑁𝑃 ≤ 𝑀𝑃 , since 𝑀𝑃 is an amply cofinitely weak

supplemented module, then by Lemma 1.3.11,

32

𝑀𝑃 𝑁𝑃

is an amply cofinitely weak

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

𝑀

supplemented module, that is ( )𝑃 is an amply cofinitely weak supplemented 𝑁

module. To show that

𝑀

is an amply cofinitely weak supplemented module, it

𝑁

𝐾

is enough to show that cofinite submodule of 𝑀𝑃

submodule of 𝐾

𝐾𝑃

𝑁

𝑁𝑃

so ( )𝑃 = 𝐾𝑃 𝑁𝑃

∩

𝐿𝑃 𝑁𝑃

≪

𝑁𝑃

𝑁

𝑀 𝑁

and as

has ample weak supplements in

𝑁𝑃

𝑁

𝑀𝑃 𝑁𝑃

. Since

𝐾

𝐾𝑃

𝑁

𝑁𝑃

, so we can easily show that ( )𝑃 =

𝐾 𝑁

is a

is a cofinite

is an amply cofinitely weak supplemented module,

has a weak supplement, say

𝑀𝑃

𝑀

𝐿𝑃 𝑁𝑃

𝐿

𝐾𝑃

𝑁

𝑁𝑃

= ( )𝑃 , where

+

𝐾

𝐿

𝑀

𝐾

𝐿

𝑀

𝑁

𝑁

𝑁

𝑁

𝑁

𝑁

𝐿𝑃 𝑁𝑃

=

𝑀𝑃 𝑁𝑃

and

, which gives ( + )𝑃 = ( )𝑃 and ( ⋂ )𝑃 ≪ ( )𝑃 . Then, by

using the given condition of the lemma, we get 𝐾

𝐿

𝑀

𝐿

𝑁

𝑁

𝑁

𝑁

1.3.4, we get ( ⋂ ) ≪ , so that

𝐾 𝑁

+

𝐿 𝑁

=

𝑀 𝑁

and by Proposition

is a weak supplement of

𝐾 𝑁

𝑀

𝑀

𝑁

𝑁

in . Hence,

is an amply cofinitely weak supplemented module. Now, in the next Lemma, we give a condition to transform the property of amply cofinitely weak supplementedness from the localization of a given module to the module itself. Lemma 2.1.13 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃, If 𝑀𝑃 is an amply cofinitely weak supplemented 𝑅𝑃 −module, then 𝑀 is an amply cofinitely weak supplemented module.

33

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. Let 𝑈 be a cofinite submodule of 𝑀, then 𝑀𝑃 𝑈𝑃

𝑀 𝑈

is finitely generated, this gives

𝑀

= ( )𝑃 is finitely generated. Thus, 𝑈𝑃 is a cofinite submodule of 𝑀𝑃 and 𝑈

as 𝑀𝑃 is amply cofinitely weak supplemented module, then 𝑈𝑃 has weak supplements in 𝑀𝑃 , so let 𝐾𝑃 be a weak supplement of 𝑈𝑃 in 𝑀𝑃 , then 𝑈𝑃 + 𝐾𝑃 = 𝑀𝑃 and 𝑈𝑃 ⋂𝐾𝑃 ≪ 𝑀𝑃 . By the same technique as in Lemma 2.1.12, we can prove that 𝐾 is a weak supplement of 𝑈 in 𝑀. Hence, 𝑀 is an amply cofinitely weak supplemented module. Now, we prove that under a certain condition the directed summand of a module which is the localization of which is an amply cofinitely weak supplemented module is also amply cofinitely weak supplemented. Corollary 2.1.14 Let 𝑀 be an 𝑅 −module and 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑀𝑃 is an amply cofinitely weak supplemented 𝑅𝑃 −module, then every direct summand of 𝑀 is an amply cofinitely weak supplemented submodule. Proof. Let 𝑈 be any direct summand of 𝑀. Since 𝑀𝑃 is an amply cofinitely weak supplemented 𝑅𝑃 −module, by Lemma 2.1.13, we get 𝑀 is an amply cofinitely weak supplemented module and then by Corollary 1.3.14, we get 𝑈 which is an amply cofinitely weak supplemented submodule.

34

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

In Corollary 1.3.14, it is proved that a homomorphic image of an amply cofinitely weak supplemented module is an amply cofinitely weak supplemented. In the following result, we droop the property of amply cofinitely weak supplementedness from the module and we give it to the localization of module and we prove that under a certain condition the result remain true. Corollary 2.1.15 Let 𝑀 and 𝑀′ be 𝑅 −modules and 𝑃 be a prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝐾: 𝑝 = 𝐾 for all 𝑝 ∉ 𝑃. If 𝑓: 𝑀 → 𝑀′ is homomorphism and 𝑀𝑃 is an amply cofinitely weak supplemented 𝑅𝑃 −module, then 𝑓(𝑀) is an amply cofinitely weak supplemented module. Proof. By Lemma 2.1.13, we have 𝑀 which is an amply cofinitely weak supplemented 𝑅 −module and then by Corollary 1.3.14, we get 𝑓(𝑀) which is an amply cofinitely weak supplemented module. 2.2 Rad- Supplemented Modules In this section, the effect of localization on Rad-supplemented modules and weak Rad-supplemented modules are studied. Several conditions are given under which certain properties of such types of modules are preserved under localization.

35

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Lemma 2.2.1 Let 𝑀 be an 𝑅 −module and 𝑃 a prime ideal of 𝑅. if 𝑆(𝐾) ⊆ 𝑃, for every proper submodule 𝐾 of 𝑀, then 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 . Proof. If 𝐾 is not a proper submodule of 𝑀, then 𝐾 = 𝑀. But we have 𝑆(𝑀) = ∅ (trivial), since if 𝑆(𝑀) ≠ ∅, so there exists 𝑟 ∈ 𝑆(𝑀), so that there exists 𝑥 ∈ 𝑀 − 𝐾 such that 𝑟𝑥 ∈ 𝑀 (Definition of 𝑆(𝑀) is 𝑆(𝑀) = {𝑟 ∈ 𝑅: 𝑟𝑥 ∈ 𝑀, for some 𝑥 ∈ 𝑀 − 𝑀}), this means 𝑥 ∈ ∅

and 𝑟𝑥 ∈ 𝑀 which is a

contradiction, so that 𝑆(𝑀) = ∅, so that trivially we get 𝑆(𝑀) = ∅ ⊆ 𝑃. Hence we get that 𝑆(𝐾) ⊆ 𝑃, for every submodule 𝐾 of 𝑀. Now, if we let 𝑆 = 𝑅 − 𝑃, then clearly 𝑆 is a multiplicative system from which we get 𝑃 = 𝑅 − 𝑆, so we have 𝑆(𝐾) ⊆ 𝑅 − 𝑆, for every submodule 𝐾 of 𝑀, this gives 𝑆(𝐾)⋂𝑆 ⊆ (𝑅 − 𝑆)⋂𝑆 = ∅, for every submodule 𝐾 of 𝑀 and as 𝑀 is a submodule of 𝑀, so by Lemma 1.3.31, we get 𝑅𝑎𝑑(𝑀𝑆 ) = (𝑅𝑎𝑑𝑀)𝑆 , that is 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 (since 𝑆 = 𝑅 − 𝑃). Lemma 2.2.2 Let 𝑀 be an 𝑅-module with submodules 𝑈 and 𝑉of 𝑀 and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑉𝑃 is a weak Rad-supplement submodule of 𝑈𝑃 in 𝑀𝑃 , then for a submodule 𝐿′ of 𝑈𝑃 , there exists a submodule 𝐿 of 𝑈 such that Rad-supplement submodule of

𝑈 𝐿

in

𝑀 𝐿

. with 𝐿′ = 𝐿𝑃 .

36

𝑉+𝐿 𝐿

is a weak

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. Since 𝑈 and 𝑉 are submodules of 𝑀 and 𝑉𝑃 is a weak Rad-supplement submodule of 𝑈𝑃 in 𝑀𝑃 , then 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ) this implies that (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ) , by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 , then by proposition 1.3.3, we get 𝑈 + 𝑉 = 𝑀 and 𝑈 ∩ 𝑉 ≤ 𝑅𝑎𝑑𝑀, hence 𝑉 is a weak Rad-supplement submodule of 𝑈 in 𝑀. Since 𝐿′ ≤ 𝑈𝑃 , then by Lemma 1.3.2 there exists a submodule 𝐿 ≤ 𝑈 such that 𝐿′ = 𝐿𝑃 . Thus by Lemma 1.3.22, submodule of

𝑈 𝐿

𝑉+𝐿 𝐿

is a weak Rad-supplement

𝑀

in . 𝐿

In Proposition 1.3.30, it is proved that every factor module of completely weak Rad-supplemented module is also a completely weak Rad-supplemented module. Now, we prove this result by replacing the module by its localization. Proposition 2.2.3 Let 𝑀 be an 𝑅-module with a submodule 𝐿 of 𝑀 and 𝑃 be any prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a completely weak Rad-supplemented 𝑅𝑃 -module, then

𝑀 𝐿

is a completely

weak Rad-supplemented 𝑅-module. Proof. Let

𝐾 𝐿

𝑀

𝐾𝑃

𝐿

𝐿𝑃

≤ . Then

≤

𝑀𝑃 𝐿𝑃

. And let

𝑈𝑃 𝐿𝑃

≤

𝐾𝑃 𝐿𝑃

, then 𝑈𝑃 ≤ 𝐾𝑃 . Since 𝑀𝑃 is a

completely weak Rad-supplemented 𝑅𝑃 -module, there exists a submodule 𝑉𝑃 of 𝐾𝑃 which holds 𝑈𝑃 + 𝑉𝑃 = 𝐾𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤ 𝑅𝑎𝑑(𝐾𝑃 ). 𝑉𝑃 is a weak Rad-

37

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

supplement of 𝑈𝑃 in 𝐾𝑃 and 𝐿𝑃 ≤ 𝑈𝑃 . Then by Lemma 2.2.2, we get that is a weak Rad-supplement of module. Therefore,

𝑀 𝐿

𝑈 𝐿

𝐾

𝐾

𝐿

𝐿

in . Hence,

𝑉+𝐿 𝐿

is weakly Rad-supplemented

is a completely weak Rad-supplemented 𝑅-module.

In the next result, we prove that under a certain condition, if the localization of a module is completely weak Rad-supplemented, then the module itself is so. Proposition 2.2.4 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a completely weak Rad-supplemented 𝑅𝑃 -module, then 𝑀 is also a completely weak Radsupplemented 𝑅-module. Proof. Let 𝐾 be any proper submodule of 𝑀 and 𝑈 be a submodule of 𝐾. So by Lemma 1.3.7, 𝐾𝑃 is a proper submodule of a completely weak Radsupplemented𝑅𝑃 −module 𝑀𝑃 and 𝑈𝑃 ≤ 𝐾𝑃 . Then there exists a submodule 𝑉𝑃 of 𝐾𝑃 , where 𝑉 ≤ 𝐾 such that 𝑈𝑃 + 𝑉𝑃 = 𝐾𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤ 𝑅𝑎𝑑(𝐾𝑃 ), since 𝐾 is a proper submodule of 𝑀 and 𝑆(𝐾) ⊆ 𝑃 so by part (12) of Corollary 1.3.6, 𝑅𝑎𝑑(𝐾𝑃 ) = (𝑅𝑎𝑑𝐾)𝑃 , and hence by Corollary 1.3.5 we have

(𝑈 + 𝑉)𝑃 = 𝑈𝑃 + 𝑉𝑃 = 𝐾𝑃

and (𝑈 ∩ 𝑉)𝑃 = 𝑈𝑃 ∩ 𝑉𝑃 ≤ (𝑅𝑎𝑑𝐾)𝑃 .

Since 𝑃 is a maximal ideal of 𝑅, so by proposition 1.3.3 and part (10) of Corollary 1.3.6, 𝑈 + 𝑉 = 𝐾 and 𝑈 ∩ 𝑉 ≤ 𝑅𝑎𝑑𝐾. Hence 𝐾is a weak Rad-

38

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

supplemented 𝑅 −module. Therefor, 𝑀 is a completely weak Radsupplemented 𝑅-module. Now, we give the following corollary to the Proposition 2.2.4. Corollary 2.2.5 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑀 = 𝑁⨁𝐿 be a direct sum of submodules 𝑁 and 𝐿 of 𝑀. If 𝑀𝑃 is a completely weak Rad-supplemented 𝑅𝑃 -module, then 𝑁 is also a completely weak Rad-supplemented 𝑅-module. Proof. The proof follows directly from Proposition 2.2.4. Now, we give a condition under which we can extend the result of Lemma1.3.29, in to the localized modules. Lemma 2.2.6 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let

𝑀 = 𝑈 + 𝑉 for

submodules 𝑈 and 𝑉 of 𝑀. If 𝑉𝑃 contains a Rad-supplement submodule of 𝑈𝑃 in 𝑀𝑃 , then 𝑈 ∩ 𝑉 has a Rad-supplement submodule in 𝑉. Proof. Let 𝐾 be a submodule of 𝑉 and suppose that a submodule 𝐾𝑃 of 𝑉𝑃 is a Rad-supplement of 𝑈𝑃 in 𝑀𝑃 . Then, we have 𝑈𝑃 + 𝐾𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝐾𝑃 ≤ (𝑅𝑎𝑑 𝐾𝑃 ), from the modular law we have 𝑈𝑃 ∩ 𝑉𝑃 + 𝐾𝑃 = 𝑉𝑃 . Since 𝐾𝑃 ≤ 𝑉𝑃 ,

39

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

then (𝑈𝑃 ∩ 𝑉𝑃 ) ∩ 𝐾𝑃 = 𝑈𝑃 ∩ 𝐾𝑃 ≤ (𝑅𝑎𝑑𝐾𝑃 ) , by Corollary 1.3.5, we get ((𝑈 ∩ 𝑉) + 𝐾)𝑃 = 𝑉𝑃 and ((𝑈 ∩ 𝑉) ∩ 𝐾)𝑃 ≤ (𝑅𝑎𝑑𝐾𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝐾𝑃 ) = (𝑅𝑎𝑑𝐾)𝑃 , hence by proposition 1.3.3, we get that (𝑈 ∩ 𝑉) + 𝐾 = 𝑉 and (𝑈 ∩ 𝑉) ∩ 𝐾 ≤ 𝑅𝑎𝑑𝐾. Thus,

𝐾 is a Rad-supplement

submodule of (𝑈 ∩ 𝑉) in 𝑉. Now, we give a condition under which we can extend the result of Lemma 1.3.19, in to the localized modules. Corollary 2.2.7 Let 𝑀 be an 𝑅-module and 𝑉 be a Rad-supplement submodule of 𝑈 in 𝑀. Let 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If (𝑈 ∩ 𝑉)𝑃 is a supplement submodule in 𝑈𝑃 , then 𝑉 is a supplement submodule in 𝑀. Proof. Let 𝐾 be a submodule of 𝑈 and (𝑈 ∩ 𝑉)𝑃 is a supplement submodule of 𝐾𝑃 in 𝑈𝑃 . Then, we have 𝐾𝑃 + (𝑈 ∩ 𝑉)𝑃 = 𝑈𝑃 and 𝐾𝑃 ∩ (𝑈 ∩ 𝑉)𝑃 ≪ (𝑈 ∩ 𝑉)𝑃 , so by Corollary 1.3.5, we get (𝐾 + (𝑈 ∩ 𝑉))𝑃 = 𝑈𝑃 and (𝐾 ∩ (𝑈 ∩ 𝑉))𝑃 ≪ (𝑈 ∩ 𝑉)𝑃 , hence by Proposition 1.3.3 and Corollary 1.3.6 we get 𝐾 + (𝑈 ∩ 𝑉) = 𝑈 and 𝐾 ∩ (𝑈 ∩ 𝑉) ≪ (𝑈 ∩ 𝑉). It follows that (𝑈 ∩ 𝑉) is supplement submodule of 𝐾 in 𝑈. Thus, by Lemma 1.3.19, we get that 𝑉 is a supplement submodule in 𝑀. In Proposition 1.3.26(2), it is proved that if every Rad-supplement submodule of a module is a Rad-supplemented module, then the module itself

40

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

is a supplemented 𝑅-module and now we extend this fact to the localized module. Proposition 2.2.8 Let 𝑀 be a reduced 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If every Radsupplement submodule of 𝑀𝑃 is a Rad-supplemented 𝑅𝑃 − module, then 𝑀 is a supplemented 𝑅-module. Proof. Let 𝑈 and 𝑉 be submodules of 𝑀 and let 𝑉𝑃 be a Rad-supplemented submodule of 𝑈𝑃 in 𝑀𝑃 . Then, we have 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤ 𝑅𝑎𝑑𝑉𝑃 . Since 𝑀 is a reduced 𝑅-module, then by part (2) of Corollary 1.3.6 we get 𝑀𝑃 is a reduced 𝑅𝑃 -module. Since 𝑉𝑃 is a Rad-supplemented submodule, then by Proposition 1.3.26(1), we get 𝑅𝑎𝑑(𝑉𝑃 ) ≪ 𝑉𝑃 , hence 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤ 𝑅𝑎𝑑𝑉𝑃 ≪ 𝑉𝑃 , then by Corollary 1.3.5, we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑉𝑃 , hence by Proposition 1.3.3 and by part (10) of Corollary 1.3.6 we get 𝑈 + 𝑉 = 𝑀 and 𝑈 ∩ 𝑉 ≪ 𝑉, it follows that 𝑉 is a supplement submodule of 𝑈 in 𝑀. Thus, 𝑀 is a supplement 𝑅-module. In the following result, we prove that if a module is a sum of two of its submodules for which the localization of one of them is supplemented, then the submodule contains a supplement of the other submodule.

41

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Corollary 2.2.9 Let 𝑀 be an 𝑅-module with submodules 𝑈 and 𝑉of 𝑀, and let 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃, suppose that 𝑀 = 𝑈 + 𝑉. If 𝑉𝑃 is a supplemented 𝑅𝑃 -module then 𝑉 contains a supplement submodule of 𝑈 in 𝑀. Proof. Let 𝐿 be a submodule of 𝑉 and let 𝐿𝑃 be a supplement of 𝑈𝑃 ∩ 𝑉𝑃 in 𝑉𝑃 . Then, we have 𝐿𝑃 + (𝑈𝑃 ∩ 𝑉𝑃 ) = 𝑉𝑃 and 𝐿𝑃 ∩ (𝑈𝑃 ∩ 𝑉𝑃 ) ≪ 𝐿𝑃 , here 𝑈𝑃 ∩ 𝐿𝑃 = 𝐿𝑃 ∩ (𝑈𝑃 ∩ 𝑉𝑃 ) ≪ 𝐿𝑃 , hence by Corollary 1.3.5, we get (𝐿 + (𝑈 + 𝑉))𝑃 = 𝑉𝑃 and (𝑈 ∩ 𝐿)𝑃 = (𝐿 ∩ (𝑈 ∩ 𝑉))𝑃 ≪ 𝐿𝑃 , by Proposition 1.3.3 and by part (10) of Corollary 1.3.6, we get 𝐿 + (𝑈 ∩ 𝑉) = 𝑉 and 𝑈 ∩ 𝐿 = 𝐿 ∩ (𝑈 ∩ 𝑉) ≪ 𝐿, this means that 𝐿 is a supplement of (𝑈 ∩ 𝑉) in 𝑉. Now 𝑀 = 𝑈 + 𝑉 = 𝑈 + (𝑈 ∩ 𝑉) + 𝐿 = 𝑈 + 𝐿 , hence we get 𝑀 = 𝑈 + 𝐿 and 𝑈 ∩ 𝐿 ≪ 𝐿 , it follows that 𝐿 is a supplement of 𝑈 in 𝑀. Thus, 𝑉 contains a supplement of 𝑈 in 𝑀. In Proposition 1.3.28, it is proved that if a module is amply Radsupplemented, then it is a hollow radical module and now we give a condition under which we can extend this result to the localized module. Proposition 2.2.10 Let 𝑅 be a Noetherian ring and 𝑀 be a simply radical 𝑅-module. Let 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is an amply Rad-supplemented 𝑅𝑃 -module, then 𝑀 is a hollow radical 𝑅-module. 42

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Proof. Let 𝑈 be a submodule of 𝑀 and suppose that 𝑈𝑃 + 𝑉 ′ = 𝑀𝑃 for a submodule 𝑉 ′ of 𝑀𝑃 , by Lemma 1.3.2 there exists a submodule 𝑉 of 𝑀 such that 𝑉 ′ = 𝑉𝑃 , hence we get 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 , By hypothesis, there exists a submodule 𝐿′ of 𝑉𝑃 such that 𝑈𝑃 + 𝐿′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝐿′ ≤ 𝑅𝑎𝑑 (𝐿′ ) , again by Lemma 1.3.2

there exists a submodule 𝐿 of 𝑉 such that 𝐿′ = 𝐿𝑃 ,

hence 𝑈𝑃 + 𝐿𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝐿𝑃 ≤ 𝑅𝑎𝑑 (𝐿𝑃 ) , by Corollary 1.3.5, we get (𝑈 + 𝐿)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝐿)𝑃 ≤ 𝑅𝑎𝑑(𝐿𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝐿𝑃 ) = (𝑅𝑎𝑑𝐿)𝑃 , hence by Proposition 1.3.3, we get that 𝑀 = 𝑈 + 𝐿 and 𝑈 ∩ 𝐿 ≤ 𝑅𝑎𝑑(𝐿). Since 𝑀 is simply radical, it follows that 𝑅𝑎𝑑(𝐿) = 𝐿 ∩ 𝑅𝑎𝑑(𝑀) = 𝐿 ∩ 𝑀 = 𝐿, so 𝐿 is radical. Therefore, 𝐿 = 𝑀 and 𝑉 = 𝑀. Hence, we deduce that 𝑈 is a small submodule in 𝑀. Thus, 𝑀 is hollow radical 𝑅 −module. Next, we prove that if every submodule of a localized module is Radsupplemented, then the module itself is Rad-supplemented. Proposition 2.2.11 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀 we have 𝑆(𝐾) ⊆ 𝑃. If every submodule of 𝑀𝑃 is a Rad-supplemented 𝑅𝑃 -module, then 𝑀 is an amply Rad-supplemented 𝑅module. Proof. Let 𝑁 be a submodule of 𝑀 and 𝐿′ ≤ 𝑀𝑃 . Then, 𝑁𝑃 ≤ 𝑀𝑃 and by Lemma 1.3.2 there exists submodules 𝐿 of 𝑀 such that 𝐿′ = 𝐿𝑃 , suppose 43

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

that 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 , by assumption there exists a submodule 𝐻′ of 𝐿𝑃 such that (𝑁𝑃 ∩ 𝐿𝑃 ) + 𝐻′ = 𝐿𝑃 and (𝑁𝑃 ∩ 𝐿𝑃 ) ∩ 𝐻′ = 𝑁𝑃 ∩ 𝐻′ ≤ Rad𝐻′ , again by Lemma 1.3.2

there exists a submodules 𝐻 ≤ 𝐿 such that 𝐻′ = 𝐻𝑃 ,

hence (𝑁𝑃 ∩ 𝐿𝑃 ) + 𝐻𝑃 = 𝐿𝑃

and

(𝑁𝑃 ∩ 𝐿𝑃 ) ∩ 𝐻𝑃 = 𝑁𝑃 ∩ 𝐻𝑃 ≤ Rad(𝐻𝑃 ).

Thus, 𝐿𝑃 = 𝐻𝑃 + (𝑁𝑃 ∩ 𝐿𝑃 ) ≤ 𝑁𝑃 + 𝐻𝑃 and hence 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 ≤ 𝑁𝑃 + 𝐻𝑃 . Therefore, 𝑀𝑃 = 𝑁𝑃 + 𝐻𝑃 and 𝑁𝑃 ∩ 𝐻𝑃 ≤ Rad(𝐻𝑃 ). By Lemma 2.2.1, we have 𝑅𝑎𝑑(𝐻𝑃 ) = (𝑅𝑎𝑑𝐻)𝑃 , hence by Corollary 1.3.5, 𝑀𝑃 = (𝑁 + 𝐻)𝑃 and (𝑁 ∩ 𝐻)𝑃 ≤ (Rad𝐻)𝑃 , and by Proposition 1.3.3, we get that 𝑀 = 𝑁 + 𝐻 and 𝑁 ∩ 𝐻 = 𝑅𝑎𝑑𝐻. Hence, 𝑁 has a Rad-supplement 𝐻 ≤ 𝐿. Thus, 𝑀 is an amply Rad-supplemented module. In Proposition 1.3.15, it is proved that if a module is a sum of two Radsupplemented submodules, then the module itself is Rad-supplemented. Now, we extend this result to the localized module. Corollary 2.2.12 Let 𝑁 and 𝐿 be Rad-supplemented 𝑅 −modules and 𝑃 be any maximal ideal of 𝑅. If 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 , then 𝑀 is a Rad−supplemented 𝑅 −module. Proof. Since 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 , then by Corollary 1.3.5, 𝑀𝑃 = (𝑁 + 𝐿)𝑃 and also by Proposition 1.3.3, we get 𝑀 = 𝑁 + 𝐿. Hence, by Proposition 1.3.15 we get that 𝑀 is a Rad-supplemented module. In Proposition 1.3.15, it is proved that a Rad-supplemented module which has zero Jacobson radical is semi simple. Now, we prove this result for the localized module. 44

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

Corollary 2.2.13 Let 𝑀 be a Rad-supplemented 𝑅 −module with a submodule 𝐿 of 𝑀 and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀 we have 𝑆(𝐾) ⊆ 𝑃. If 𝐿𝑃 ∩ 𝑅𝑎𝑑𝑀𝑃 = 0, then 𝐿 is semi simple. Proof. Since 𝐿𝑃 ∩ 𝑅𝑎𝑑𝑀𝑃 = 0 and by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 , then by Corollary 1.3.5 we get(𝐿 ∩ 𝑅𝑎𝑑𝑀)𝑃 = 0, and by Proposition 1.3.3, we get 𝐿 ∩ 𝑅𝑎𝑑𝑀 = 0, hence by Proposition 1.3.15, we get that 𝐿 is semi simple. In Proposition 1.3.16, it is proved that every supplement submodule of a weak Rad-supplemented module is also weak Rad-supplemented. Now, we prove this result for the localized module. Proposition 2.2.14 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a weak Radsupplemented 𝑅-module, then every supplement submodule of 𝑀 is a weak Rad-supplemented 𝑅-module. Proof. Let 𝐾 be a supplement submodule of 𝑀. For any submodule 𝑁 of , since 𝑀𝑃 is a weak Rad-supplemented module, then there exists 𝐿′ ≤ 𝑀𝑃 such that 𝑀𝑃 = 𝑁𝑃 + 𝐿′ and 𝑁𝑃 ∩ 𝐿′ ≤ 𝑅𝑎𝑑𝑀𝑃 , by Lemma 1.3.2

there exists a

submodule 𝐿 of 𝑀 such that 𝐿′ = 𝐿𝑃 , hence we get 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 and 𝑁𝑃 ∩ 45

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

𝐿𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ). Thus, 𝐾𝑃 = 𝐾𝑃 ∩ 𝑀𝑃 = 𝐾𝑃 ∩ (𝑁𝑃 + 𝐿𝑃 ) = 𝑁𝑃 + (𝐾𝑃 ∩ 𝐿𝑃 ) and 𝑁𝑃 ∩ (𝐾𝑃 ∩ 𝐿𝑃 ) = 𝐾𝑃 ∩ (𝑁𝑃 ∩ 𝐿𝑃 ) ≤ 𝐾𝑃 ∩ 𝑅𝑎𝑑(𝑀𝑃 ) = 𝑅𝑎𝑑(𝐾𝑃 ), Lemma

1.3.21.

Hence,

𝑁𝑃 + (𝐾𝑃 ∩ 𝐿𝑃 ) = 𝐾𝑃

by

and 𝑁𝑃 ∩ (𝐾𝑃 ∩ 𝐿𝑃 ) ≤

𝑅𝑎𝑑(𝐾𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝐾𝑃 ) = (𝑅𝑎𝑑𝐾)𝑃 , hence by Corollary

1.3.5

we

get

(𝑁 + (𝐾 ∩ 𝐿))𝑃 = 𝐾𝑃

and(𝑁 ∩ (𝐾 ∩ 𝐿))𝑃 ≤

(𝑅𝑎𝑑𝐾)𝑃 , and hence by Proposition 1.3.3, we get that 𝑁 + (𝐾 ∩ 𝐿) = 𝐾 and 𝑁 ∩ (𝐾 ∩ 𝐿) ≤ 𝑅𝑎𝑑𝐾. Therefore, 𝐾 is a weak Rad-supplemented 𝑅module. Next, we prove that, if the sum of the localization of two submodules of a module has a Rad-supplement submodule and if one of the submodules is Rad-supplemented, then the other submodule has a Rad-supplement submodule. Proposition 2.2.15 Let 𝑀 be an 𝑅-module with submodules 𝑈, 𝑉 ≤ 𝑀, let 𝑈 be a Radsupplemented module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀 we have 𝑆(𝐾) ⊆ 𝑃. If 𝑈𝑃 + 𝑉𝑃 has a Radsupplement submodule in 𝑀𝑃 , then so 𝑉 has a Rad-supplement submodule in 𝑀. Proof. Let 𝐿 be a submodule of 𝑉. Since 𝑈𝑃 + 𝑉𝑃 has a Rad-supplement in 𝑀𝑃 , suppose that 𝐿𝑃 is a Rad-supplement of 𝑈𝑃 + 𝑉𝑃 , hence we get 𝐿𝑃 + (𝑈𝑃 + 𝑉𝑃 ) = 𝑀𝑃 and 𝐿𝑃 ∩ (𝑈𝑃 + 𝑉𝑃 ) ≤ 𝑅𝑎𝑑(𝐿𝑃 ) , by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝐿𝑃 ) = (𝑅𝑎𝑑𝐿)𝑃 , hence by Corollary 1.3.5, we get (𝐿 + (𝑈 + 46

Chapter Two

Amply Weak Supplemented Modules and Rad- Supplemented Modules

𝑉))𝑃 = 𝑀𝑃 and(𝐿 ∩ (𝑈 + 𝑉))𝑃 ≤ (𝑅𝑎𝑑𝐿)𝑃 , and by Proposition 1.3.3, we get that 𝐿 + (𝑈 + 𝑉) = 𝑀 and 𝐿 ∩ (𝑈 + 𝑉) ≤ 𝑅𝑎𝑑(𝐿). For (𝐿 + 𝑉) ∩ 𝑈 , since 𝑈 is a Rad-supplemented module, there exists 𝐾 ≤ 𝑈 such that (𝐿 + 𝑉) ∩ 𝑈 + 𝐾 = 𝑈 and (𝐿 + 𝑉) ∩ 𝐾 ≤ 𝑅𝑎𝑑𝐾. Thus we have 𝐿 + 𝑉 + 𝐾 = 𝑀 and (𝐿 + 𝑉) ∩ 𝐾 ≤ 𝑅𝑎𝑑𝐾 , that is 𝐾 is a Rad-supplement of 𝐿 + 𝑉 in 𝑀. It is clear that (𝐿 + 𝐾) + 𝑉 = 𝑀 , since 𝐾 + 𝑉 ≤ 𝑈 + 𝑉 , 𝐿 ∩ (𝐾 + 𝑉) ≤ 𝐿 ∩ (𝑈 + 𝑉) ≤ 𝑅𝑎𝑑𝐿 , we get that (𝐿 + 𝐾) ∩ 𝑉 ≤ 𝐿 ∩ (𝐾 + 𝑉) + 𝐾 ∩ (𝐿 + 𝑉) ≤ 𝑅𝑎𝑑𝐿 + 𝑅𝑎𝑑𝐾 ≤ 𝑅𝑎𝑑(𝐿 + 𝐾). Hence we get (𝐿 + 𝐾) + 𝑉 = 𝑀 and (𝐿 + 𝐾) ∩ 𝑉 ≤ 𝑅𝑎𝑑(𝐿 + 𝐾),

it

means

that 𝐿 + 𝐾

is

a

Rad-supplemented

submodule of 𝑉 in 𝑀. Thus, 𝑉 has a Rad-supplement submodule in 𝑀.

47

Chapter Three Small Submodule, Semisimple Submodule and Some Types of Supplemented Modules

Chapter Three

small, semi simple submodule and some types of supplemented Modules

Chapter Three Small Submodule, Semisimple Submodule and Some Types of Supplemented Modules This chapter consists of two sections. In the first section, some results concerning the localization of the modules are proved. It also studies the effect of localization on certain types of modules such as small submodule, semi simple submodules and some other types. In the second section, some further results concerning the localization of the modules are proved. It also studies the effect of localization on certain types of modules such as supplement submodules, wsupplemented modules and weak Rad-supplemented modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization.

3.1 Small submodule and Semi Simple Submodule. Lemma 3.1.1 Let 𝑀 be an 𝑅 −module and 𝑃 be any prime ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑈, 𝑉 be submodules of 𝑀 with 𝑈 ≤ 𝑉 ≤ 𝑀. Then (1) If 𝑈𝑃 ≪ 𝑀𝑃 , then any submodule of 𝑈 is also small in 𝑀. (2) If 𝑈𝑃 ≪ 𝑉𝑃 , then 𝑈 ≪ 𝑀. (3) If 𝑈𝑃 ≪ 𝑀𝑃 and 𝑈𝑃 is a supplement in 𝑀𝑃 , then 𝑈 ≪ 𝑉. Proof. The proof of follows directly from Corollary 1.3.6 and Lemma 1.3.8.

48

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small, semi simple submodule and some types of supplemented Modules

It is known that, if two submodules of a module are small in the module, then the sum of them is also small in the module and conversely. Now, we extend this result to the localization of the submodules. Proposition 3.1.2 Let 𝑀 be an 𝑅 −module with submodules 𝑈 and 𝑉 of 𝑀 and 𝑃 be any prime ideal 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑈𝑃 ≪ 𝑀𝑃 and 𝑉𝑃 ≪ 𝑀𝑃 , then 𝑈 + 𝑉 ≪ 𝑀. Proof. Let (𝑈 + 𝑉) + 𝐿 = 𝑀 for a submodule 𝐿 of 𝑀, since 𝑈𝑃 ≪ 𝑀𝑃 and 𝑉𝑃 ≪ 𝑀𝑃 then by Corollary 1.3.6 we get 𝑈 ≪ 𝑀 and 𝑉 ≪ 𝑀, now since (𝑈 + 𝑉) + 𝐿 = 𝑈 + (𝑉 + 𝐿) = 𝑀 and 𝑈 ≪ 𝑀, then 𝑉 + 𝐿 = 𝑀, and since 𝑉 ≪ 𝑀, hence we get 𝐿 = 𝑀. Thus, 𝑈 + 𝑉 ≪ 𝑀. Proposition 3.1.3 Let 𝑀 be an 𝑅 −module with submodules 𝑈 and 𝑉 of 𝑀 and 𝑃 be any prime ideal of 𝑅 such that, for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑈𝑃 + 𝑉𝑃 ≪ 𝑀𝑃 , then 𝑈 ≪ 𝑀 and 𝑉 ≪ 𝑀. Proof. Since 𝑈𝑃 ≤ (𝑈𝑃 + 𝑉𝑃 ) then by Lemma 1.3.2 we get 𝑈 ≤ 𝑈 + 𝑉, and by Lemma 3.1.1(1), we get 𝑈 ≪ 𝑀, in the same way we get 𝑉 ≪ 𝑀. Next, we prove that the simplness property of modules is a localization property.

49

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small, semi simple submodule and some types of supplemented Modules

Proposition 3.1.4 Let 𝑀 be an 𝑅 −module with a submodule 𝑁 of 𝑀 and 𝑃 any prime ideal of 𝑅. Then, 𝑁 is a simple submodule of 𝑀 if and only if 𝑁𝑃 is a simple submodule of 𝑀𝑃 . Proof. Let

𝑁 be a simple submodule of 𝑀. Then there exists a no zero proper

submodule 𝐾 of 𝑁, by Lemma 1.3.2 we get 𝐾𝑃 is a proper submodule of 𝑁𝑃 . To prove 𝐾𝑃 ≠ 0, let 𝐾𝑃 = 0, then by simple calculation we get 𝐾 = 0, which is contradiction. Hence 𝐾𝑃 is a nonzero proper submodule of 𝑁𝑃 . Thus 𝑁𝑃 is a simple submodule of 𝑀𝑃 . Conversely, let 𝑁𝑃 is a simple submodule of 𝑀𝑃 . Then there exists a no zero proper submodule 𝐾𝑃 of 𝑁𝑃 , by Lemma 1.3.2 we get 𝐾 is a proper submodule of 𝑁. To prove 𝐾 ≠ 0, let 𝐾 = 0, then by simple calculation we get 𝐾𝑃 = 0, which is contradiction. Hence 𝐾 is a nonzero proper submodule of 𝑁. Thus 𝑁 is a simple submodule of 𝑀. Next, we prove that under certain condition the semisimplness property of modules is a localization property. Proposition 3.1.5 Let 𝑀 be an 𝑅 −module with a submodule 𝑁 of 𝑀 and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Then, 𝑁 is a semi simple submodule of 𝑀 if and only if 𝑁𝑃 is a semi simple submodule of 𝑀𝑃 .

50

Chapter Three

small, semi simple submodule and some types of supplemented Modules

Proof. Let 𝑁 be a semi simple submodule of 𝑀 and 𝑉 ′ be a submodule of 𝑁𝑃 . Then, by Lemma 1.3.2 there exists a submodule 𝑉 ≤ 𝑁 such that 𝑉 ′ = 𝑉𝑃 , since 𝑉 ≤ 𝑁 and 𝑁 is a semi simple submodule of 𝑀, then there exists a submodule 𝑈 of 𝑁 such that 𝑈⨁𝑉 = 𝑁 , this means that 𝑁 = 𝑉 + 𝑈 and 𝑉 ∩ 𝑈 = {0}, hence by Corollary 1.3.5 and Proposition 1.3.3 we get 𝑁𝑃 = (𝑈 + 𝑉)𝑃 = 𝑉𝑃 + 𝐿𝑃 and (𝑉 ∩ 𝐿)𝑃 = 𝑉𝑃 ∩ 𝐿𝑃 = {0}, that is, 𝑁𝑃 = 𝑉𝑃 ⨁𝐿𝑃 . Thus, 𝑁𝑃 is a semi simple submodule of 𝑀𝑃 . Conversely, let 𝑉 be a submodule of 𝑁. Since 𝑁𝑃 is a semi simple submodule of 𝑀𝑃 , then 𝑉𝑃 is a submodule of 𝑁𝑃 , so there exists a submodule 𝑈 ′ of 𝑁𝑃 such that 𝑁𝑃 = 𝑉𝑃 ⨁𝑈 ′, by Lemma 1.3.2 there exists a submodule 𝑈 ≤ 𝑁 such that 𝑈 ′ = 𝑈𝑃 , hence 𝑁𝑃 = 𝑉𝑃 ⨁𝑈𝑃 , that is 𝑁𝑃 = 𝑉𝑃 + 𝑈𝑃 and 𝑉𝑃 ∩ 𝑈𝑃 = {0}, by Corollary

1.3.5

we

get (𝑈 + 𝑉)𝑃 = 𝑁𝑃

and (𝑈 ∩ 𝑉)𝑃 = {0},

hence

by

Proposition 1.3.3 we get 𝑈 + 𝑉 = 𝑁 and 𝑈 ∩ 𝑉 = {0}, this means that 𝑈⨁𝑉 = 𝑁. Thus, 𝑁 is a semi simple submodule of 𝑀. In Lemma 1.3.17, it is proved that, if a module is a sum of two submodules one of them is semi simple, then the semi simple submodule can be replaced by a submodule of it which has zero intersection with the other submodules. Corollary 3.1.6 Let 𝑀 be an 𝑅 −module with submodules 𝑁 and 𝐿 of 𝑀 and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 where 𝑁𝑃 is a semi simple of 𝑀𝑃 , then 𝑀 = 𝑈⨁𝐿 for some submodule 𝑈 of 𝑁.

51

Chapter Three

small, semi simple submodule and some types of supplemented Modules

Proof. Since 𝑀𝑃 = 𝑁𝑃 + 𝐿𝑃 , then by Corollary 1.3.5 𝑀𝑃 = (𝑁 + 𝐿)𝑃 and by Proposition 1.3.3 𝑀 = 𝑁 + 𝐿, and since 𝑁𝑃 is a semi simple of 𝑀𝑃 , then by Proposition 3.1.5 𝑁 is also semi simple of 𝑀. Thus, by Lemma 1.3.17we get that 𝑀 = 𝑈⨁𝐿 for some submodule 𝑈 of 𝑁. In the next result, we give a condition under which the property of lying above submodules is a localization property. Theorem 3.1.7 Let 𝑀 be an 𝑅 −module with submodules 𝑈 and 𝑉 of 𝑀, and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Then, a submodule 𝑈𝑃 of 𝑀𝑃 lies above a submodule 𝑉𝑃 of

𝑀𝑃 if and only if a

submodule 𝑈 of 𝑀 lies above a submodule 𝑉 of 𝑀. Proof. Since a submodule 𝑈𝑃 of 𝑀𝑃 lies above a submodule 𝑉𝑃 of 𝑀𝑃 , then we have 𝑈𝑃

𝑉𝑃 ≤ 𝑈𝑃 and 𝑉 ≤ 𝑈 and

𝑈 𝑉

≪

𝑉𝑃

𝑀𝑃 𝑉𝑃

, by Proposition 1.3.3 and Corollary 1.3.6, we get that

𝑀

≪ , this means that a submodule 𝑈 of 𝑀 lies above a submodule 𝑉 𝑉

of 𝑀. Conversely, since a submodule 𝑈 of 𝑀 lies above a submodule 𝑉 of 𝑀, then we have 𝑉 ≤ 𝑈 and 𝑉𝑃 ≤ 𝑈𝑃 and,

𝑈𝑃 𝑉𝑃

𝑈 𝑉

𝑀

≪ , by Proposition 1.3.3 and Corollary 1.3.6, we get that

≪

𝑉

𝑀𝑃 𝑉𝑃

this means that a submodule 𝑈𝑃 of 𝑀𝑃 lies above a

submodule 𝑉𝑃 of 𝑀𝑃 .

52

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small, semi simple submodule and some types of supplemented Modules

3.2 Weak Rad- Supplemented Modules and Some Other Types of Supplemented Modules In Proposition 1.3.16, it is proved that every factor module of a weak Radsupplemented module is also a weak Rad-supplemented. Now, we extend this result to the localized modules. Proposition 3.2.1 Let 𝑀 be an 𝑅-module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a weak Rad-supplemented 𝑅 −module, then every factor module of 𝑀 is a weak Rad-supplemented 𝑅 −module. Proof. Let 𝑁 be any submodule of 𝑀 and

𝐿 𝑁

be any submodule of

𝑀 𝑁

for 𝐿 ≤ 𝑀.

Since 𝑀𝑃 is a weak Rad-supplemented 𝑅 −module, then there exists 𝐾 ′ ≤ 𝑀𝑃 such that 𝐿𝑃 + 𝐾 ′ = 𝑀𝑃 and 𝐿𝑃 ∩ 𝐾 ′ ≤ 𝑅𝑎𝑑(𝑀𝑃 ) , by Lemma 1.3.2 there exists a submodule 𝐾 ≤ 𝑀 such that 𝐾 ′ = 𝐾𝑃 , hence we get 𝑀𝑃 = 𝐾𝑃 + 𝐿𝑃 and 𝐾𝑃 ∩ 𝐿𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Corollary 1.3.5 (𝐾 + 𝐿)𝑃 = 𝑀𝑃 and (𝐾 ∩ 𝐿)𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 , hence by Proposition 1.3.3 we get that 𝑀 = 𝐾 + 𝐿 and 𝐾 ∩ 𝐿 ≤ 𝑅𝑎𝑑(𝑀). Thus function 𝑓: 𝑀 →

𝑀 𝑁

𝑀 𝑁

𝑁 𝐿 𝑁

∩

that

𝑁 𝑀 𝑁

𝑀

= 𝑓(𝐿 ∩ 𝐾) ≤ 𝑓(𝑅𝑎𝑑(𝑀)) ≤ 𝑅𝑎𝑑 ( ). 𝑁

𝐾+𝑁

𝐿 𝑁

+

𝐾+𝑁 𝑁

. Let the

be canonical epimorphism. Since 𝐾 ∩ 𝐿 ≤ 𝑅𝑎𝑑(𝑀), as the

same argument as in Proposition 1.3.16, we can get (𝑁+(𝐾∩𝐿)

=

𝑀

𝐿

𝑁

𝑁

≤ 𝑅𝑎𝑑( ), this means that

Hence,

𝐿 𝑁

∩ 𝐿 𝑁

𝐾+𝑁 𝑁

+

=

𝐾+𝑁 𝑁

(𝐿∩(𝐾+𝑁) 𝑁

=

𝑀 𝑁

=

and

is a weak Rad-supplemented submodule, so

is a weak Rad-supplemented 𝑅 −module. 53

Chapter Three

small, semi simple submodule and some types of supplemented Modules

Next, we generalize the result in Proposition 1.3.34, to the localized modules. Lemma 3.2.2 Let 𝑀 be an 𝑅 −module with submodules 𝑈 and 𝑉of 𝑀 and 𝑈 be a weak Radsupplemented 𝑅 −module. Let 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑈𝑃 + 𝑉𝑃 has a weak Radsupplement in 𝑀𝑃 , then so does 𝑉. Proof. By assumption there exists a submodule 𝑁 ′of 𝑀𝑃 such that (𝑈𝑃 + 𝑉𝑃 ) + 𝑁 ′ = 𝑀𝑃 and

(𝑈𝑃 + 𝑉𝑃 ) ∩ 𝑁 ′ ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 1.3.2 there exists a

submodule 𝑁 ≤ 𝑀 such that 𝑁 ′ = 𝑁𝑃 , hence we get (𝑈𝑃 + 𝑉𝑃 ) + 𝑁𝑃 = 𝑀𝑃 and (𝑈𝑃 + 𝑉𝑃 ) ∩ 𝑁𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ) , by Corollary 1.3.5, (𝑈 + 𝑉) + 𝑁)𝑃 = 𝑀𝑃 and (𝑈 + 𝑉) ∩ 𝑁)𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ),

by

Lemma

2.2.1,

we

have 𝑅𝑎𝑑(𝑀𝑃 ) =

(𝑅𝑎𝑑𝑀)𝑃 , then by Proposition 1.3.3 we get (𝑈 + 𝑉) + 𝑁 = 𝑀 and (𝑈 + 𝑉) ∩ 𝑁 ≤ 𝑅𝑎𝑑𝑀, since 𝑈 is a weak Rad-supplemented module, there exists a submodule 𝐿 ≤ 𝑈 such that 𝑈 ∩ (𝑁 + 𝑉) + 𝐿 = 𝑈 and (𝑁 + 𝑉) ∩ 𝐿 ≤ 𝑅𝑎𝑑(𝑈) . Thus, (𝑁 + 𝐿) + 𝑉 = 𝑀

and

𝑈 ∩ (𝑁 + 𝐿) ≤ (𝑈 + 𝑉) ∩ 𝑁 + 𝐿 ∩ (𝑁 + 𝑉) ≤

𝑅𝑎𝑑(𝑀), that means that (𝑁 + 𝐿) is a weak Rad-supplement submodule of 𝑉 in 𝑀. In the next result, we give a condition under which we can extend the result of Proposition 1.3.26(3), to the localized module. Proposition 3.2.3 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a radical 𝑅𝑃 −module, then 𝑀 is a Rad-supplemented 𝑅 −module.

54

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Proof. By assumption, for every submodule 𝑈 ′ of 𝑀𝑃 we have 𝑈 ′ + 𝑀𝑃 = 𝑀𝑃 and 𝑈 ′ ∩ 𝑀𝑃 = 𝑈 ′ ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 1.3.2 there exists a submodule 𝑈 ≤ 𝑀 such that 𝑈 ′ = 𝑈𝑃 , hence we get (𝑈𝑃 + 𝑀𝑃 ) = 𝑀𝑃 and (𝑈𝑃 ∩ 𝑀𝑃 ) ≤ 𝑅𝑎𝑑(𝑀𝑃 ) , by Corollary 1.3.5 (𝑈 + 𝑀)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑀)𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 , then by Proposition 1.3.3 we get 𝑈 + 𝑀 = 𝑀 and 𝑈 ∩ 𝑀 ≤ 𝑅𝑎𝑑𝑀, this means that 𝑀 is a Rad-supplemented 𝑅 −module. In Proposition 1.3.33, it is proved that if a module is a sum of two modules each has an ample Rad-supplement, then their intersection has also an ample Radsupplement. Now, we generalize this result to the localized module. Proposition 3.2.4 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑀 = 𝑈 + 𝑉 , for submodules 𝑈 and 𝑉 of 𝑀. If 𝑈𝑃 and 𝑉𝑃 have ample Rad-supplements in 𝑀𝑃 , then 𝑈 ∩ 𝑉 also has ample Rad-supplements in 𝑀. Proof. Let 𝑁 be a submodule of 𝑀 and 𝑈 ∩ 𝑉 + 𝑁 = 𝑀. Then 𝑈 = 𝑀 ∩ 𝑈 = (𝑈 ∩ 𝑉 + 𝑁) ∩ 𝑈 and as 𝑈 ∩ 𝑉 ≤ 𝑈, by modular law we get 𝑈 = 𝑈 ∩ 𝑉 + (𝑁 ∩ 𝑈) and since 𝑈 ∩ 𝑉 ≤ 𝑉, so by the same way we can get 𝑉 = 𝑈 ∩ 𝑉 + (𝑁 ∩ 𝑉). Also, by using modular law we have, 𝑀 = 𝑈 + 𝑁 ∩ 𝑉 and 𝑀 = 𝑉 + 𝑁 ∩ 𝑈. Since, 𝑈𝑃 and 𝑉𝑃 have ample Rad-supplements in 𝑀𝑃 , there exists 𝑇 ′ ≤ 𝑁𝑃 ∩ 𝑉𝑃 and 𝐿′ ≤ 𝑁𝑃 ∩ 𝑈𝑃 such that 𝑈𝑃 + 𝐿′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝐿′ ≤ 𝑅𝑎𝑑(𝐿′ ), also 𝑉𝑃 + 𝑇 ′ = 𝑀𝑃 and 𝑉𝑃 ∩ 𝑇 ′ ≤ 𝑅𝑎𝑑(𝑇 ′ ), by Lemma 1.3.2 there exists submodules 𝐿 ≤ 𝑁 ∩ 𝑈 and 𝑇 ≤ 𝑁 ∩ 𝑉 such that 𝐿′ = 𝐿𝑃 and 𝑇 ′ = 𝑇𝑃 , hence we get that 𝑈𝑃 + 𝐿𝑃 = 𝑀𝑃 55

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and 𝑈𝑃 ∩ 𝐿𝑃 ≤ 𝑅𝑎𝑑(𝐿𝑃 ), also 𝑉𝑃 + 𝑇𝑃 = 𝑀𝑃 and 𝑉𝑃 ∩ 𝑇𝑃 ≤ 𝑅𝑎𝑑(𝑇𝑃 ). Thus 𝐿𝑃 + 𝑇𝑃 ≤ 𝑁𝑃 and 𝑈𝑃 = 𝑈𝑃 ∩ 𝑉𝑃 + 𝐿𝑃 and 𝑉𝑃 = 𝑈𝑃 ∩ 𝑉𝑃 + 𝑇𝑃 , therefore (𝑈𝑃 ∩ 𝑉𝑃 ) + (𝑇𝑃 + 𝐿𝑃 ) = 𝑀𝑃 𝑅𝑎𝑑(𝐿𝑃 + 𝑇𝑃 ),

and hence

(𝑈𝑃 ∩ 𝑉𝑃 ) ∩ (𝑇𝑃 + 𝐿𝑃 ) = (𝑉𝑃 ∩ 𝐿𝑃 ) + (𝑈𝑃 ∩ 𝑇𝑃 ) ≤ (𝑈𝑃 ∩ 𝑉𝑃 ) + (𝑇𝑃 + 𝐿𝑃 ) = 𝑀𝑃

(𝑇𝑃 + 𝐿𝑃 ) ≤ 𝑅𝑎𝑑(𝐿𝑃 + 𝑇𝑃 ), by Corollary 1.3.5

and (𝑈𝑃 ∩ 𝑉𝑃 ) ∩

we get that [(𝑈 ∩ 𝑉) +

(𝐿 + 𝑇)]𝑃 = 𝑀𝑃 and [(𝑈 ∩ 𝑉) ∩ (𝐿 + 𝑇)]𝑃 ≤ 𝑅𝑎𝑑((𝐿 + 𝑇)𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑((𝐿 + 𝑇)𝑃 ) = (𝑅𝑎𝑑(𝐿 + 𝑇))𝑃 , by Proposition 1.3.3 we get that (𝑈 ∩ 𝑉) + (𝐿 + 𝑇) = 𝑀

and (𝑈 ∩ 𝑉) ∩ (𝐿 + 𝑇) ≤ 𝑅𝑎𝑑(𝐿 + 𝑇),

this

means

that (𝑈 ∩ 𝑉) has ample Rad-supplements in 𝑀. In Proposition 1.3.32, it is proved that a finitely generated and weak Radsupplemented module is a weakly supplemented module and now we give a condition under which we can extend this result to the localized module. Proposition 3.2.5 Let 𝑀 be a finitely generated 𝑅 −module and 𝑃 be any maximal ideal of 𝑅, such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is weak Radsupplemented 𝑅𝑃 −module, then 𝑀 is a weakly supplemented 𝑅 −module. Proof. Let 𝑈 be a submodule of 𝑀. Since 𝑀𝑃 is a weak Rad-supplemented 𝑅𝑃 −module, then 𝑈𝑃 is a submodule of 𝑀𝑃 , so there exists a submodule 𝑉 ′ of 𝑀𝑃 such that 𝑈𝑃 + 𝑉 ′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉 ′ ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 1.3.2 there exists a submodule 𝑉 ≤ 𝑀 such that 𝑉 ′ = 𝑉𝑃 , hence

𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≤

𝑅𝑎𝑑(𝑀𝑃 ), by Corollary 1.3.5 we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≤ 𝑅𝑎𝑑(𝑀𝑃 ), by Lemma 2.2.1, we have 𝑅𝑎𝑑(𝑀𝑃 ) = (𝑅𝑎𝑑𝑀)𝑃 , hence by Proposition 1.3.3 we get that 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≤ 𝑅𝑎𝑑(𝑀), and since 𝑀 is a finitely 56

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generated 𝑅 −module then, 𝑅𝑎𝑑(𝑀) ≪ 𝑀. Thus, 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑀, this means that 𝑀 is weakly supplemented 𝑅 −module. In the following result, we prove that under certain condition the 𝑤 −supplementedness property of modules is a localization property. Lemma 3.2.6 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀 we have 𝑆(𝐾) ⊆ 𝑃. Then, 𝑀𝑃 is a 𝑤 −supplemented 𝑅𝑃 −module if and only if 𝑀 is a 𝑤 −supplemented 𝑅 −module. Proof. Let 𝑈 be a semi simple submodule of 𝑀. Then by Proposition 3.1.5 𝑈𝑃 is a semi simple submodule of 𝑀𝑃 , and since 𝑀𝑃 is a 𝑤 −supplemented 𝑅𝑃 −module, then for a submodule 𝑈𝑃 of 𝑀𝑃 there exists a submodule 𝑉 ′ of 𝑀𝑃 such that 𝑈𝑃 + 𝑉 ′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉 ′ ≪ 𝑉 ′ , by Lemma 1.3.2 there exists a submodule 𝑉 ≤ 𝑀 such that 𝑉 ′ = 𝑉𝑃 , hence 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≪ 𝑉𝑃 , by Corollary 1.3.5 we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑉𝑃 , hence by Proposition 1.3.3

and

Corollary 1.3.6 we get that 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑉, that is 𝑉 is a supplement submodule of 𝑈 in 𝑀, this means that 𝑀 is a 𝑤 −supplemented 𝑅 −module. Conversely, let 𝑈 ′ be a semi simple submodule of 𝑀𝑃 . Then by Lemma 1.3.2 there exists a submodule 𝑈 ≤ 𝑀 such that 𝑈 ′ = 𝑈𝑃 , by Proposition 3.1.5 𝑈 is a semi simple submodule of 𝑀, since 𝑀 is 𝑤 −supplemented 𝑅 −module then there exists a submodule 𝑉 of 𝑀 such that 𝑈 + 𝑉 = 𝑀 and 𝑈 ∩ 𝑉 ≪ 𝑉, hence by Proposition 1.3.3 and Corollary 1.3.6 we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑉𝑃 , by Corollary 1.3.5 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≪ 𝑉𝑃 , that is, 𝑉𝑃 is a

57

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supplement submodule of 𝑈𝑃 in 𝑀𝑃 , this means that 𝑀𝑃 is a 𝑤 −supplemented 𝑅𝑃 −module. In Proposition 1.3.18, it is proved that every semi simple submodule of a 𝑤 −supplemented module has a supplement that is a direct summand. We extend this result to the localized module. Corollary 3.2.7 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀 we have 𝑆(𝐾) ⊆ 𝑃. If 𝑀𝑃 is a 𝑤 −supplemented 𝑅𝑃 −module, then every semi simple submodule of 𝑀 has a supplement which is a direct summand. Proof. The proof is clear by Lemma 3.2.6 and Proposition 1.3.18. In the next two results, we give a condition under which we can extend the result of Lemma 1.3.27, in to the localized module. Proposition 3.2.8 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑉 be a supplement submodule of 𝑈 in 𝑀, 𝑁 and 𝑇 be submodules of 𝑉. If 𝑇𝑃 is a supplement submodule of 𝑁𝑃 in 𝑉𝑃 , then 𝑇 is a supplement submodule of 𝑈 + 𝑁 in 𝑀. Proof. We have 𝑉𝑃 = 𝑁𝑃 + 𝑇𝑃 and 𝑁𝑃 ∩ 𝑇𝑃 ≪ 𝑇𝑃 , by Corollary 1.3.5, we have 𝑉𝑃 = (𝑁 + 𝑇)𝑃 and (𝑁 ∩ 𝑇)𝑃 ≪ 𝑇𝑃 and by Proposition 1.3.3 and Corollary 58

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1.3.6, we get 𝑉 = 𝑁 + 𝑇 and 𝑁 ∩ 𝑇 ≪ 𝑇, this means that 𝑇 is a supplement submodule of 𝑈 in 𝑉. Now, let 𝑈 + 𝑁 + 𝐿 = 𝑀 for 𝐿 ≤ 𝑇. Then 𝐿 + 𝑁 ≤ 𝑉 and since 𝑉 is a supplement submodule of 𝑈 in 𝑀, then by Lemma 1.3.20 , we get that 𝑇 is a supplement submodule of 𝑈 + 𝑁 in 𝑀. Proposition 3.2.9 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑉 be a supplement submodule of 𝑈 in 𝑀, 𝑁 and 𝑇 be submodules of 𝑉. If 𝑇𝑃 is a supplement submodule of (𝑈 + 𝑁)𝑃 in 𝑀𝑃 , then 𝑇 is a supplement submodule of 𝑁 in 𝑉. Proof. We have 𝑀𝑃 = (𝑈 + 𝑁)𝑃 + 𝑇𝑃 and (𝑈 + 𝑁)𝑃 ∩ 𝑇𝑃 ≪ 𝑇𝑃 , by Corollary 1.3.5 𝑀𝑃 = (𝑁 + 𝑈 + 𝑇)𝑃 and ((𝑁 + 𝑈) ∩ 𝑇)𝑃 ≪ 𝑇𝑃 , by Proposition 1.3.3

and

Corollary 1.3.6 we get 𝑀 = 𝑁 + 𝑈 + 𝑇 and ( 𝑁 + 𝑈) ∩ 𝑇 ≪ 𝑇. This means that 𝑇 is a supplement submodule of 𝑈 + 𝑁 in 𝑀, so by Lemma 1.3.20, 𝑇 is a supplement submodule of 𝑁 in 𝑉. In the next result, we give a condition under which we can generalize the result in Corollary 1.3.26, to the localized modules. Lemma 3.2.10 Let 𝑀 be an 𝑅 −module and 𝑃 be a maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let 𝑈 and 𝑉 be mutual supplement submodule in 𝑀. If for 𝑆 ≤ 𝑈, 𝐿′ is a supplement submodule of 𝑆𝑃 in 𝑈𝑃 and for 𝑁 ≤ 𝑉, 𝑇 ′ is a supplement submodule of 𝑁𝑃 in 𝑉𝑃 , then 𝐿 + 𝑇 is a supplement submodule of 𝑁 + 𝑆 in 𝑀. 59

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Proof. Since 𝐿′ is a supplement submodule of 𝑆𝑃 in 𝑈𝑃 and 𝑇 ′ is a supplement submodule of 𝑁𝑃 in 𝑉𝑃 , then by Lemma 1.3.2 there exist submodules 𝐿 ≤ 𝑈 and 𝑇 ≤ 𝑉 such that 𝐿′ = 𝐿𝑃 and 𝑇 ′ = 𝑇𝑃 , we have 𝑈𝑃 = 𝑆𝑃 + 𝐿𝑃 and 𝑆𝑃 ∩ 𝐿𝑃 ≪ 𝐿𝑃 , also 𝑉𝑃 = 𝑁𝑃 + 𝑇𝑃 and 𝑁𝑃 ∩ 𝑇𝑃 ≪ 𝑇𝑃 , then by Corollary 1.3.5, we have 𝑈𝑃 = (𝑆 + 𝐿)𝑃 and (𝑆 ∩ 𝐿)𝑃 ≪ 𝐿𝑃 , also 𝑉𝑃 = (𝑁 + 𝑇)𝑃 and (𝑁 ∩ 𝑇)𝑃 ≪ 𝑇𝑃 . Also, by Proposition 1.3.3 and Corollary 1.3.6, we get that 𝑈 = 𝑆 + 𝐿 and 𝑆 ∩ 𝐿 ≪ 𝐿. Also 𝑉 = 𝑁 + 𝑇 and 𝑁 ∩ 𝑇 ≪ 𝑇. Now, since 𝑈 = 𝑆 + 𝐿 and 𝑉 is a supplement submodule of 𝑈 in 𝑀, then by Lemma 1.3.20 , we get that 𝑇 is a supplement submodule of (𝑆 + 𝐿 + 𝑁) in 𝑀 and (𝑆 + 𝐿 + 𝑁) ∩ 𝑇 ≪ 𝑇, also since 𝑉 = 𝑁 + 𝑇 and 𝑈 is a supplement submodule of 𝑉 in 𝑀, then by Lemma 1.3.20 , we get that 𝐿 is a supplement submodule of (𝑆 + 𝑇 + 𝑁) in 𝑀 and (𝑆 + 𝑇 + 𝑁) ∩ 𝐿 ≪ 𝐿. Now, we have 𝑀 = 𝑈 + 𝑉 = (𝑆 + 𝐿) + (𝑁 + 𝑇) = (𝑆 + 𝑁) + (𝐿 + 𝑇) and (𝑆 + 𝑁) ∩ (𝐿 + 𝑇) ≤ 𝐿 ∩ (𝑆 + 𝑁 + 𝑇) + 𝑇 ∩ (𝑆 + 𝑁 + 𝐿) ≪ 𝐿 + 𝑇. Hence, 𝑀 = (𝑆 + 𝑁) + (𝐿 + 𝑇) and (𝑆 + 𝑁) ∩ (𝐿 + 𝑇) ≪ (𝐿 + 𝑇), this means that (𝐿 + 𝑇) is a supplement submodule of (𝑁 + 𝑆) in 𝑀. In the last two results, we extend the results in Theorem 1.3.23 and Theorem 1.3.24, to the localized module. Proposition 3.2.11 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let a submodule 𝑈 of 𝑀 lie above a submodule 𝐿 of 𝑀. If 𝑈𝑃 and 𝐿𝑃 have weak supplements in 𝑀𝑃 , then 𝑈 and 𝐿 have the same weak supplements in 𝑀.

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Proof. Let 𝑉 ′ be a weak supplement submodule of 𝑈𝑃 in 𝑀𝑃 . Than, 𝑈𝑃 + 𝑉 ′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉 ′ ≪ 𝑀𝑃 , by Lemma 1.3.2 there exists a submodule 𝑉 ≤ 𝑀 such that 𝑉 ′ = 𝑉𝑃 , hence 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≪ 𝑀𝑃 , by Corollary 1.3.5 we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑀𝑃 , hence by Proposition 1.3.3

and

Corollary 1.3.6 we get that 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑀, that is 𝑉 is a weak supplement submodule of 𝑈 in 𝑀. Now, since 𝐿 ≤ 𝑈, by Lemma 1.3.13 , 𝐿 + 𝑉 = 𝑀 and since 𝑉 is a weak supplement submodule of 𝑈 in 𝑀 and 𝐿 ≤ 𝑈, then 𝐿 ∩ 𝑉 ≤ 𝑈 ∩ 𝑉 ≪ 𝑀. Thus, 𝑉 is a weak supplement submodule of 𝐿 in 𝑀. Now, let 𝑇 ′ be a weak supplement submodule of 𝐿𝑃 in 𝑀𝑃 , Than, 𝐿𝑃 + 𝑇 ′ = 𝑀𝑃 and 𝐿𝑃 ∩ 𝑇 ′ ≪ 𝑀𝑃 , by Lemma 1.3.2 there exists a submodule 𝑇 ≤ 𝑀 such that 𝑇 ′ = 𝑇𝑃 , hence 𝐿𝑃 + 𝑇𝑃 = 𝑀𝑃 and 𝐿𝑃 ∩ 𝑇𝑃 ≪ 𝑀𝑃 , by Corollary 1.3.5 we get (𝐿 + 𝑇)𝑃 = 𝑀𝑃 and (𝐿 ∩ 𝑇)𝑃 ≪ 𝑀𝑃 , hence by Proposition 1.3.3

and

Corollary 1.3.6, we get that 𝑀 = 𝐿 + 𝑇 and 𝐿 ∩ 𝑇 ≪ 𝑀, that is 𝑇 is a weak supplement submodule of 𝐿 in 𝑀. Hence, by Theorem 1.3.24, we obtains that 𝑇 is a weak supplement submodule of 𝑈 in 𝑀. Proposition 3.2.12 Let 𝑀 be an 𝑅 −module and 𝑃 be any maximal ideal of 𝑅 such that for each proper submodule 𝐾 of 𝑀, we have 𝑆(𝐾) ⊆ 𝑃. Let the submodule 𝑈 of 𝑀 lies above the submodule 𝐿 of 𝑀. If 𝑈𝑃 and 𝐿𝑃 have supplements in 𝑀𝑃 , then 𝑈 and 𝐿 have the same supplements in 𝑀. Proof. Let 𝑉 ′ be a supplement submodule of 𝑈𝑃 in 𝑀𝑃 . Than, 𝑈𝑃 + 𝑉 ′ = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉 ′ ≪ 𝑉 ′ , by Lemma 1.3.2 there exists a submodule 𝑉 ≤ 𝑀 such that 𝑉 ′ = 𝑉𝑃 , 61

Chapter Three

small, semi simple submodule and some types of supplemented Modules

hence 𝑈𝑃 + 𝑉𝑃 = 𝑀𝑃 and 𝑈𝑃 ∩ 𝑉𝑃 ≪ 𝑉𝑃 , by Corollary 1.3.5, we get (𝑈 + 𝑉)𝑃 = 𝑀𝑃 and (𝑈 ∩ 𝑉)𝑃 ≪ 𝑉𝑃 , hence by Proposition 1.3.3 and Corollary 1.3.6, we get that 𝑀 = 𝑈 + 𝑉 and 𝑈 ∩ 𝑉 ≪ 𝑉, that is 𝑉 is a supplement submodule of 𝑈 in 𝑀. Now, since 𝐿 ≤ 𝑈, by Lemma 1.3.13 , 𝐿 + 𝑉 = 𝑀 and since 𝑉 is a supplement submodule of 𝑈 in 𝑀 and 𝐿 ≤ 𝑈, then 𝐿 ∩ 𝑉 ≤ 𝑈 ∩ 𝑉 ≪ 𝑉. Thus, 𝑉 is a supplement submodule of 𝐿 in 𝑀. Now let 𝑇 ′ be a supplement submodule of 𝐿𝑃 in 𝑀𝑃 , Than, 𝐿𝑃 + 𝑇 ′ = 𝑀𝑃 and 𝐿𝑃 ∩ 𝑇 ′ ≪ 𝑇 ′, by Lemma 1.3.2 there exists a submodule 𝑇 ≤ 𝑀 such that 𝑇 ′ = 𝑇𝑃 , hence 𝐿𝑃 + 𝑇𝑃 = 𝑀𝑃 and 𝐿𝑃 ∩ 𝑇𝑃 ≪ 𝑇𝑃 , by Corollary 1.3.5, we get (𝐿 + 𝑇)𝑃 = 𝑀𝑃 and (𝐿 ∩ 𝑇)𝑃 ≪ 𝑇𝑃 , hence by Proposition 1.3.3 and Corollary 1.3.6, we get that 𝑀 = 𝐿 + 𝑇 and 𝐿 ∩ 𝑇 ≪ 𝑇, that is, 𝑇 is a supplement submodule of 𝐿 in 𝑀. Hence, by Theorem 1.3.23, we get that 𝑇 is a supplement submodule of 𝑈 in 𝑀.

62

References

References [1] Akray, I., Jabbar, A., and Sazeedeh, R. (2012), ”Some Conditions Under Which Certain Types of Modules Possess Localization Property”, Journal Duhok university, Vol.15, No. 1, 144-152. [2] Atiiyah, M., and Macdonald, I. (1969), ”Introduction to Commutative Algebra”, Addison-Wesley Publishing Company. [3] Bilhan, G. (2005), ”Amply FWS modules”, Journal of Arts and Sciences, Sayi 4, 31-36. [4] Burcu, N., Turkmen, E. and Pancar, A. (2009), “Completely Weak Radsupplemented Modules”, International Journal of Computational Cognttion, Vol. 7, No. 2, 48-50. [5] Choubey, S., Pandeya B., and Gupta, A. (2012), ”Amply Weak Radsupplemented Modules”, International Journal of Algebra, Vol. 6, No. 27, 1335 – 1341. [6] Clark, J., Lomp, C., Vajana, N. and Wisbauer, R. (2006), “Lifting Modules”, Birkhauser Verlag Basel, Boston-berlin. [7] Gokhan B. and Ayse T. (2013), “A variation of Supplemented Modules”, Turkish Journal Of Mathematics, 37, 418-426. [8] Jabbar, A., and Hasan, N. (2016), “On Locally Multiplication Modules”, International Mathematical Forum, Vol. 11, No. 5, 213-226. [9] Jabbar, A. (2013), “A Generalization of Prime and Weakly Prime submodules”, Pure Mathematical sciences, Vol. 2, No. 1, 1-11. [10] Larsen, M. and McCarthy, P. (1971), “Multiplicative Theory of Ideals”, Academic Press, New York and London. [11] Nebiyev, C. and Pancar, A. (2013), “On Supplement Submodules”, ISSN, Vol. 65, No. 7, 1027-3190.

63

References

[12] Nebiyev, C. (2005), “Amply weak supplemented modules”, International Journal Computational Cogintion, Vol. 3, No. 1, 88-90. [13] Talebi, Y. and Mahmoudi, A. (2011), “On Rad-⊕-supplemented modules”, Thai Journal of Mathematics, Volume 9, No. 2, 367–375. [14] Talebi, Y., Hamzecolaei, A., and Tutuncu, D. (2009), ”On Rad+supplemented modules”, Hadronic Journal, 32, 505-512. [15] Turkmen, E. and Pancar, A. (2009), “On Radical Supplemented Modules”, International Journal of Computational Cognition, Vol. 7, No. 1, 62-64. [16] Turkmen, E. and Pancar, A. (2011), “Some properties of Rad-supplemented modules”, International Journal of the Physical Sciences Vol. 6, No. 35, 7904 7909. [17] Turkmen, E. and Pancar, A. (2012), “Characterizations of Rad-supplemented modules”, Miskolc Mathematical Notes, Vol. 13, No. 2, pp. 569 – 580. [18] Wisbauer, R. (1991), “Foundations of Module and Ring Theory”, Gordon and Breach Science Publishers. [19] Yongduo, W. and Nanqing, D. (2006), “Generalized Supplemented Modules”, Taiwanese Journal of Mathematics, Vol. 10, No. 6, 1589-1601.

64

‫اخلالصة‬

‫اخلالصة‬ ‫فى هذا العمل متت دراسة بعض البنى اجلربية مثل املقاسات اجلزئية الصغرية‪ ,‬املقاسات اجلزئية التكميلية‪ ,‬املقاسات‬ ‫اجلزئية شبه البسيطة ‪ ,‬املقاسات اجلزئية التكميلية من النمط (𝒘) ‪ ,‬املقاسات اجلزئية الفوقية ‪ ,‬املقاسات التكميلية‬ ‫الضعيفة الفسيحة ‪ ,‬املقاسات التكميلية الضعيفة منتهية التكميل الفسيحة ‪ ,‬املقاسات التكميلية جذريا ‪ ,‬املقاسات‬ ‫التكميلية جذريا ضعيفا حيث متت الربهنة على نتائج عديدة تتعلق بهذه البنى اجلربية وكذلك مت حتديد بعض‬ ‫العالقات التى تربط هذه البنى ببعضها‪.‬‬ ‫ان اهلدف الرئيسى من هذه الرسالة هو اجياد خصائص لبعض انواع من املقاسات والتى تبقى ثابتة حتت تأثري عملية‬ ‫التموضع وكذلك تهدف اىل توسيع بعض اخلواص اجلربيةاملعروفة من املقاسات اىل متوضعاتها‪.‬‬ ‫باالضافة اىل ما ذكر اعاله لقد مت احلصول على شروط عديدة والتى جتعل من متوضع مقاس تكميلى ضعيف فسيح‬ ‫مقاسا تكميليا ضعيفا فسيحا واعطيت عدة شروط اخرى والذى ميكننا كل واحد منها من نقل بعض خواص متوضع املقاس‬ ‫اىل املقاس نفسه ‪ ,‬على سبيل املثال اعطينا شروطا بتوفرها نتمكن من نقل خاصية التكميل الضعيف الفسيح للمقاسات‬ ‫من متوضوعاتها اىل املقاسات نفسها‪.‬‬ ‫واخريا وليس اخرا مت اجياد شروط اخرى عند توفرها تبقى خواصا حمددة للمقاسات التكميلية جذريا واملقاسات‬ ‫التكميلية جذريا ضعيفا ثابتة حتت تأثري عملية متوضع املقاسات‪.‬‬

‫بعض اخلصائص النواع حمددة من املقاسات والتى حيافظ عليهاحتت عملية التموضع‬

‫رسالة‬ ‫مقدمة اىل جملس الكلية الرتبية‬ ‫يف جامعة السليمانية‬ ‫كجزء من متطلبات نيل شهادة‬ ‫ماجستري يف علوم الرياضيات‬ ‫)اجلرب(‬

‫من قبل‬ ‫خبتيار حممود رحيم‬ ‫بكالوريوس يف الرياضيات‪ -‬جامعة السليمابية‪2012-‬‬ ‫بإشراف‬ ‫ا‪.‬د‪.‬عادل قادر جبار‬

‫شوال‪1438-‬‬

‫ثوختة‬

‫ثوختة‬ ‫لةم كارةدا ليَكوَلَينةوة لةسةر هةندىَ بونياتة جةبريةكان كرا وةك بةشة ثيَوةرة بضووكةكان ‪ ,‬بةشة ثيَوةرة‬ ‫تةواوكارييةكان ‪ ,‬بةشة ثيَوةرة نيمضة سادةكان ‪ ,‬بةشة ثيَوةرة تةواوكارييةكان لة شيَوازى ( 𝑤) ‪ ,‬بةشة ثيَوةرة‬ ‫سةرووكان ‪ ,‬ثيَوةرة تةواوكارييةالوازة فراوانةكان ‪ ,‬ثيَوةرة تةواكارييةكوَتايي هاتووةالوازة فراوانةكان ‪ ,‬ثيَوةرة‬ ‫تةواوكاريية رةطييةكان و ثيَوةرة تةواكاريية رةطيية الوازةكان و ضةند ئةجناميَك سةلَميَندران كة ثةيوةنديدارن بةم‬ ‫بونياتة جةبريانةوة و هةروةها ضةند ثةيوةنديَك ديارى كران كة ئةم بونياتة جةبريانة دةبةسنت بة يةكةوة‪.‬‬ ‫ئاماجنى سةرةكى ئةم ماستةرنامةية دوَزينةوةى تايبةمتةندى هةندىَ جوَرى ثيَوةرةكانة كة ناطوَريَن لةذيَر‬ ‫كاريطةرى بةلوَكاىل كردنى ثيَوةرةكان وةهةروةها فراوانكردنى هةندىَ تايبةمتةندية جةبرية زانراوةكان لة‬ ‫ثيَوةرةكانةوة بوَ لوَكاليكراوةكانيان‪.‬‬ ‫جطة لةوانةى سةرةوة كة بامسان ليَوةكرد ضةند مةرجيَكمان دةستكةوت كة هةر يةكةيان لوَكالَيكراوى ثيَوةريَكى‬ ‫تةواكاريي الوازةى فراوان دةكاتة ثيَوةرة يَكى تةواكاريي الوازةى فراوان وةهةروةها ضةند مةرجيَكى تر دران بة دابني‬ ‫بوونيان دةتوانني هةندىَ تايبةمتةندى لوَكاليكراوى ثيَوةريَك بطوازينةوة بوَ ثيَوةرةكة خوَى بوَ منوونة هةندىَ مةرج‬ ‫دران بة دابني بوونيان دةتوانني تايبةمتةندى تةواكاري الوازى فراوانى ثيَوةرةكان بطوازينةوة لة لوَكاليكراوةكانيان بوَ‬ ‫ثيَوةرةكان خوَيان‪.‬‬ ‫لة كوَتاييدا هةندىَ مةرجى تر دوَزرانةوة كةبة بوونيان هةندىَ تايبةمتةندى ديارى كراوى ثيَوةرة تةواوكاريية‬ ‫رةطييةكان و ثيَوةرة تةواكاريية رةطيية الوازةكان ناطوَريَن لة ذيَر كاريطةرى بةلوَكالَيكردنى ثيَوةرةكان‪.‬‬

‫هةنديَ لة تايبةمتةنديةكانى ضةند جوَريَكى دياريكراو لة ثيَوةرةكان كة‬ ‫دةثاريَزريَن لة ذيَر كاريطةرى بة خوَجىَ كردن‬ ‫نامةيةكة‬ ‫ثيَشكةش كراوة بة ئةجنومةنى كوَليَذى ثةروةردة‬ ‫لة زانكؤى سليَماني‬ ‫وةك بةشيَك لة ثيَداويستيةكانى بةدةستهيَنانى برِوانامةى‬ ‫ماستةر لةزانسيت مامتاتيك‬ ‫)جةبر(‬ ‫لةاليةن‬ ‫بةختيار مةمحوود رِةحيم‬ ‫بةكالؤريؤس لة مامتاتيك ‪ -‬زانكؤي سليَمانى‪)2012( -‬‬ ‫بة سةرثةرشيت‬ ‫ث‪.‬د‪.‬عادل قادر جبار‬

‫ثوشثةرِِِ‪2717-‬‬