Additive Representations of Elements in Rings: A Survey
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Keywords Units · Idempotents · k-good rings · von Neumann regular ring · Unit sum number · Clean rings
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2000 Mathematics Subject Classification 16U60 · 16D50
1 Additive Unit Representation
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Abstract This article presents a brief survey of the work done on various additive representations of elements in rings. In particular, we study rings where each element is a sum of units; rings where each element is a sum of idempotents; rings where each element is a sum of idempotents and units; and rings where each element is a sum of additive commutators. We have also included a number of open problems in this survey to generate further interest among readers in this topic.
The historical origin of study of the additive unit structure of rings may be traced back to the work of Dieudonné on noncommutative Galois theory [11]. In [26], Hochschild studied additive unit representations of elements in simple algebras and proved that each element of a simple algebra over any field is a sum of units. Later, Zelinsky [53] proved that the ring of linear transformations is generated additively by its unit elements. Zelinsky showed that every linear transformation of a vector space V over a division ring is the sum of two invertible linear transformations, except when V is one-dimensional over F2 , the field of two elements. Zelinsky also noted in his paper that this result follows from a previous result of Wolfson [52]. See [14] for another proof of this result. Apart from the ring of linear transformations, there are several other natural classes of rings that are generated by their unit elements. Let X be a completely regular Hausdorff space. Then every element in the ring C(X) of real-valued continuous
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A.K. Srivastava (B) Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA e-mail:
[email protected] © Springer Science+Business Media Singapore 2016 S.T. Rizvi et al. (eds.), Algebra and its Applications, Springer Proceedings in Mathematics & Statistics 174, DOI 10.1007/978-981-10-1651-6_4 419130_1_En_4_Chapter TYPESET
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Definition 1 An element x in a ring R is called a k-good element if x can be written as the sum of k units in R. We say that a ring R is a k-good ring if each element x ∈ R is a k-good element. The unit sum number of a ring R is defined as usn(R) =
⎧ ⎨ k if k is the smallest integer such that R is k-good
ω if every element of R is a sum of units but R is not k-good for any k
⎩ ∞ there exists an element a in R that cannot be written as sum of units
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functions on X is the sum of two units. For any f (x) ∈ C(X), we have f (x) = [(f (x) ∨ 0) + 1] + [(f (x) ∧ 0) − 1]. Every element in a real or complex Banach algebra B is the sum of two units. For any a ∈ B, there exists a scalar λ (= 0) such that a − λ is a unit and a = (a − λ) + λ. On the other hand, any ring having a homomorphic image isomorphic to F2 × F2 cannot be additively generated by its units because in F2 × F2 , the element (1, 0) cannot be expressed as a sum of any number of units. In this area of research, a lot of focus has been on representing ring elements as the sum of a fixed number of unit elements.
A natural question that one may think at this point is the following: given any positive integer n ≥ 2, can we construct a ring whose unit sum number is exactly n. The answer is yes and it follows from a construction of Herwig and Ziegler [25]. Although Herwig and Ziegler stated their result in a weaker form, but a careful examination of their proof reveals that they actually prove more that we they have stated.
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Theorem 2 (Herwig and Ziegler, [25]) For each n ≥ 2, there exists a domain R with usn(R) = n. The key point in proving this theorem is the observation that if R is an integral domain, n ≥ 2, an integer and x, a nonzero element of R, then R is contained in a domain S satisfying the following properties:
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(1) x is the sum of n units in S, and (2) if an element of R is the sum of k < n units in S, then it is the sum of k units in R as well.
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Theorem 3 Let R be any ring.
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(1) Any diagonal matrix over R is a 2-good element. (2) The matrix ring Mn (R) is 3-good for all n ≥ 2. In view of the above theorem, it immediately follows that if R is a ring that can be realized as a matrix ring Mn (S), n ≥ 2 over some ring S, then the only possible values of unit sum number for R are 2 and 3. Henriksen [24] gave examples of rings R for which Mn (R) is not 2-good. The example given by Henriksen was generalized by Vámos [47] in the next proposition.
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Now that we know there exists domain with any given arbitrary unit sum number, it makes sense to ask if we can construct specific class of rings with any given unit sum number. First, we consider matrix rings. The following result is due to Henriksen [24].
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Proposition 4 Let R be a ring, n ≥ 2 an integer and let L = Ra1 + · · · + Ran be a left ideal of R generated by the elements a1 , . . . , an ∈ R. Let A be the n × n matrix whose entries are all zero except for the first column which is (a1 , . . . , an )T . Suppose that (1) L cannot be generated by fewer than n elements, and (2) zero is the only 2-good element in L.
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Since for matrix rings Mn (R) the only possible values for unit sum number are 2 and 3, all the focus has been on finding class of rings R for which Mn (R) will have unit sum number 2. We give below a list of results in this direction. We say that an n × n matrix A over a ring R admits a diagonal reduction if there exist invertible matrices P, Q ∈ Mn (R) such that PAQ is a diagonal matrix. Following Ara et al. [1], a ring R is called an elementary divisor ring if every square matrix over R admits a diagonal reduction. This definition is less stringent than the one proposed by Kaplansky in [29]. The class of elementary divisor rings includes right self-injective von Neumann regular rings, unit regular rings. As we have already seen that if R is any ring, then any n × n (where n ≥ 2) diagonal matrix over R is the sum of two invertible matrices. Thus it follows that
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Example 1 If R is a commutative noetherian domain of Krull dimension greater than 1, then for any n ≥ 1 there is an n-generated ideal of R which cannot be generated by fewer than n elements. It follows that if R is any of the rings Z[x] or F[x, y], where F is a field, then condition (2) is also satisfied by any proper ideal of R. So for these rings, Mn (R) is not 2-good for all n ≥ 2. Thus, usn(Mn (R)) = 3 if R = Z[x] or F[x, y].
Lemma 5 If R is an elementary divisor ring, then Mn (R) has unit sum number 2 for n ≥ 2. A permutation matrix is a square matrix that has exactly one 1 in each row and column, and all other entries 0. An n × n matrix A = [aij ] is said to avoid a permutation matrix P = [pij ] if, for all i, j, such that pij = 1, aij = 0.
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Theorem 6 (Vámos and Wiegand [48]) Let R be any ring and n ≥ 2. If A ∈ Mn(R) avoids a permutation matrix P, then A is 2-good. Call a matrix A = [aij ] to be b-banded if for each i, j with |i − j| ≥ b, aij = 0. For example, a diagonal matrix is a 1-banded matrix. As a consequence of the above theorem, it follows that Corollary 7 (Vámos and Wiegand [48]) If A ∈ Mn(R) is a b-banded matrix with n ≥ 2b, then A is 2-good.
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Define the meeting number m(P, B) of a permutation matrix P with a matrix B as m(P, B) := |{positions i, j where both B, P are nonzero}|.
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Corollary 9 If B is an n × n matrix in block diagonal form where the blocks have size ≤ n/2, then B is 2-good. We have seen that if R is k-good, then Mn (R) is k-good. Clearly, the converse is not true. For example, M2 (F2 ) is 2-good, but F2 is not 2-good. This shows corner ring of a k-good ring need not be k-good. Also, R being k-good does not imply R[x] is k-good. In fact, usn(R[x]) = ∞ as the only units in the polynomial ring are the units in R. In case of infinite matrices too the situation is not bad. Wang and Chen [49] proved the following.
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The above proposition yields that
Theorem 10 Let R be a 2-good ring. Then the ring B(R) of all ω × ω row-andcolumn-finite matrices over R has unit sum number 2. However, if R is any arbitrary ring, then the ring B(R) has unit sum number 2 or 3. In 1958 Skornyakov asked: Is every von Neumann regular ring, which does not have a homomorphic image isomorphic to F2 × F2 , additively generated by its units? This question of Skornyakov was answered in the negative by George Bergman who constructed an example of a von Neumann regular ring in which not all elements are sums of units. Bergman’s example given below was first reported in a paper by Handelman [21].
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Proposition 8 Let B = diag(B1 , B2 , . . . , Bt ) be an n × n matrix and each Bi is a matrix of size at most n2 . Then there exists an n × n permutation matrix Q that avoids the blocks of B.
Example 2 Let k be any field, and A = k[[x]] be the power series ring in one variable over k. Let K be the field of fractions of A. Let R = {r ∈ End(Ak ) : there exists q ∈ K, a positive integer n, with r(a) = qa for all a ∈ (x n )}. Then R is a von Neumann regular ring which is not generated by its units. So the above example is an example of a von Neumann regular ring with unit sum number ∞. But, in general, we do not yet know what are all possible values for unit sum number of a von Neumann regular ring. For (von Neumann regular) right self-injective rings, the complete characterization of unit sum numbers was given by Khurana and Srivastava in [30] and [31]. If R is a right self-injective ring then R/J(R) is a von Neumann regular right self-injective ring. Since unit sum number of R is same as the unit sum number of R/J(R), in order to classify unit sum numbers of right self-injective rings, it suffices to do that for any von Neumann regular right selfinjective ring. Kaplansky developed type theory as a classification tool for certain class of rings of operators and that theory applies to von Neumann regular right self-injective rings. A careful examination of the type theory leads us to the fact that a von Neumann regular right self-injective ring R is a direct product of an abelian regular ring and proper matrix rings over elementary divisor rings. It was this crucial observation that helped Khurana and Srivastava to extend the result of Zelinsky and give complete description of unit sum number of right self-injective rings.
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Theorem 11 (Khurana and Srivastava, [30] and [31]) The unit sum number of a nonzero right self-injective ring R is 2, ω or ∞. Moreover,
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As the above theorem shows that for right self-injective von Neumann regular rings, the only possibility of unit sum numbers is 2, ω and ∞, we propose the following problem. Problem 12 Does there exist a von Neumann regular ring with unit sum number other than 2, ω and ∞?
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The following conjecture proposed in [42] still remains open. Conjecture 13 A unit regular ring R has unit sum number 2 if and only if R has no homomorphic image isomorphic to F2 . In [16] it is shown that if the identity in a ring R with stable range one is a sum of two units, then every von Neumann regular element in R is a sum of two units. Consequently, it follows that every element in a unit regular ring R is a sum of two units if the identity in R is a sum of two units. The following problem was posed by Henriksen and it is still open.
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(1) usn(R) = 2 if and only if R has no homomorphic image isomorphic to F2 . (2) usn(R) = ω if and only if R has a homomorphic image isomorphic to F2 , but has no homomorphic image isomorphic to F2 × F2 . In this case every non-invertible element of R is a sum of either two or three units. (3) usn(R) = ∞ if and only if R has a homomorphic image isomorphic to F2 × F2 .
Problem 14 (Henriksen, [24]) If R is a simple algebra over a field with more than 2 elements, and R has an idempotent e = 0 and 1, then R is generated additively by its units. Does R have a finite unit sum number? Definition 15 The unit sum number of a module M, denoted by usn(M), is the unit sum number of its endomorphism ring. In the sense of above definition, Zelinsky’s result states that usn(VD ) = 2 for a vector space over a division ring D except when V is one-dimensional over D. Laszlo Fuchs raised the question of determining when an endomorphism ring is generated additively by automorphisms. For abelian groups, this question has been studied by many authors. In [20] Hill showed that if G is a totally projective p-group with p = 2, then any endomorphism of G is the sum of two automorphisms. As a consequence, Hill also obtained that if the primary group G is a direct sum of countable groups and has an odd prime associated with it, then any endomorphism of G is the sum of two automorphisms. For a primary group G having no elements of infinite height, Stringall [45] gave necessary and sufficient conditions for endomorphism ring of G to be additively generated by its automorphisms. Khurana and Srivastava studied this question for several classes of modules in [31]. They proved the following.
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Theorem 17 If X is a class of modules closed under isomorphisms and M, an X -automorphism-invariant module with u : M → X, a monomorphic X -envelope such that End(X)/J(End(X)) is a von Neumann regular right self-injective ring and idempotents lift modulo J(End(X)). Then the unit sum number of M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 .
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As a continuous module is quasi-continuous and also satisfies the exchange property, it follows that the unit sum number of a continuous (and hence also of injective and quasi-injective) module M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 . In [31] it is also shown that the unit sum number of a flat cotorsion (in particular, pure injective) module M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 . Let M be a module and X , a class of R-modules closed under isomorphisms. In [18], a module M is called X -automorphism-invariant if there exists an X -envelope u : M → X satisfying that for any automorphism g : X → X there exists an endomorphism f : M → M such that u ◦ f = g ◦ u. Recently, Guil Asensio, Keskin Tütüncü and Srivastava [17] have shown that
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In particular, as a consequence of the above theorem, we have the following. Theorem 18 Let M be a module that is invariant under automorphisms of its injective envelope or pure-injective envelope then the unit sum number of M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 . Also, if M is a flat module that is invariant under automorphisms of its cotorsion envelope then the unit sum number of M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 .
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1.1 Additive Unit Representation of Rings of Integers of Number Fields
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Theorem 16 Let M be a quasi-continuous module with finite exchange property. Then the unit sum number of M is 2 if and only if End(M) has no homomorphic image isomorphic to F2 .
Let K = Q(ξ) be a number field (that is, a finite extension of Q) and let OK be the ring of integers of K. The following theorem due to Frei [13] shows abundance of ring of integers of number field additively generated by units. Theorem 19 For any number field K, there exists a number field L containing K, such that the ring of integers of L is generated additively by its units, that is, usn(OL ) ≤ ω.
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The following result of Jarden and Narkiewicz [28] is quite interesting as it shows that although there are so many ring of integers of number field additively generated by their units, but the ring of integers of any number field of finite degree cannot have a finite unit sum number. 419130_1_En_4_Chapter TYPESET
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As a consequence, it follows that a finitely generated integral domain of zero characteristic cannot be n-good for any n. Thus, in particular
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Theorem 21 The ring of integers of any number field of finite degree cannot have finite unit sum number. In [3] and [4], the authors give conditions under which rings of integers of various number fields are generated additively by their units. √ Theorem 22 Let K = Q( d), where d ∈ Z is square-free. Then usn(OK ) = ω if and only if
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Theorem 20 If R is a finitely generated integral domain of characteristic zero and n ≥ 1 is an integer, then there exists a constant An (R) such that every arithmetic progression in R having more than An (R) elements contains an element which is not a sum of n units.
(1) d ∈ {−1, −3} or (2) d > 0, d ≡ 1 mod 4, and d + 1 or d − 1 is a perfect square, or (3) d > 0, d ≡ 1 mod 4, and d + 4 or d − 1 is a perfect square. √ Theorem 23 Let d be a cube-free integer and K = Q( 3 d). Then usn(OK ) = ω if and only if
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1.2 Further Generalizations of Zelinsky’s Result
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Chen [10] has recently shown that if V is a countably generated right vector space over a division ring D where |D| > 3, then for each linear transformation T on VD , there exist invertible linear transformations P and Q on VD such that T − P, T − P−1 and T 2 − Q2 are invertible. Wang and Zhou continued this line of investigation in [50]. They considered the following properties
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• A ring R is said to satisfy the property (P), if for all a ∈ R, there exists a unit u ∈ R such that a + u, a − u−1 are units. • A ring R is said to satisfy the property (Q), if for all a ∈ R, there exists a unit u ∈ R such that a − u, a − u−1 are units. Wang and Zhou showed the following.
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Theorem 24 Let EndD (V ) be the ring of linear transformations of a right vector space V over a division ring D.
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Lemma 25 Let R be any ring.
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Using structure theory of von Neumann regular right self-injective rings and the above lemma, Siddique and Srivastava [40] obtained the following.
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(1) Let R be an elementary divisor ring. Then the matrix ring Mn (R) is twin-good for each n ≥ 3. (2) If R is an abelian regular ring, then the matrix ring Mn (R) is twin-good for each n ≥ 2. In particular, if D is a division ring, then the matrix ring Mn (D) is twin-good for each n ≥ 2.
Theorem 26 A right self-injective ring R is twin-good if and only if R has no homomorphic image isomorphic to F2 or F3 . As a consequence, it follows that
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Corollary 27 For any linear transformation T on a right vector space V over a division ring D, there exists an invertible linear transformation S on V such that both T − S and T + S are invertible, except when V is one-dimensional over F2 or F3 .
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Corollary 28 Let M be a quasi-continuous module with finite exchange property and R = End(M). Then R is twin-good if and only if R has no homomorphic image isomorphic to F2 or F3 . In particular, the endomorphism ring of a continuous module or a flat cotorsion (in particular, pure injective) or a Harada module is twin-good if and only if it has no homomorphic image isomorphic to F2 or F3 . We would like to raise the following problem.
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Problem 29 Let R be a Dedekind domain with finite class number c. Let n ≥ 2c. Is Mn (R) a twin-good ring? The result of Siddique and Srivastava has recently been generalized in [32] where it is shown that Theorem 30 If no field of order less than n + 2 is a homomorphic image of a right self-injective ring R, then for any element a ∈ R and central units u1 , . . . , un in R, there exists a unit u ∈ R, such that a + ui u is a unit in R for each i.
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In [40] a ring R is said to be a twin-good ring if for each x ∈ R there exists a unit u ∈ R such that both x + u and x − u are units in R. Clearly every twin-good ring is 2-good. However, there are numerous examples of 2-good rings which are not twin-good. For example, F3 is 2-good but not twin-good. Clearly, if D is a division ring such that |D| ≥ 4, then D is twin-good. Siddique and Srivastava [40] proved the following.
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In fact, in the above theorem, instead of assuming that units u1 , . . . , un are in center of ring R, it suffices to assume that the group of units U(R) is abelian. In [46] Tang and Zhou have shown that each linear transformation of a vector space V over a division ring D is a sum of two commuting invertible linear transformations if and only if V is finite-dimensional and |D| ≥ 3. Recently, it has been generalized in [39] where it is shown that if E is a -injective module such that each endomorphism of E is a sum of two commuting automorphisms then E is directly finite. We propose the following conjecture.
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Conjecture 31 Let E be an injective module. Then each endomorphism of E can be expressed as a sum of two commuting automorphisms if and only if E is directly finite and End(E) has no homomorphic image isomorphic to Mn (F2 ) for any n.
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Theorem 32 The following conditions are equivalent for a ring R;
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Hirano and Tominaga [27] studied rings in which each element is the sum of two idempotents. These rings may be seen as a generalization of Booleanrings.Let S and S 0 satisfies T be Boolean rings and M be an T -S-bimodule. Then the ring R = MT the property that each element of R is the sum of two idempotents. However, this ring R is not Boolean. For n ≥ 2, the matrix ring Mn (R) over any ring R contains an element which is not the sum of two idempotents. Hirano and Tominaga proved the following.
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(1) R is a commutative ring in which each element is the sum of two idempotents. (2) R is a ring in which each element is the sum of two commuting idempotents. (3) x 3 = x for each element x ∈ R. As a consequence of this theorem, they further deduced that
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An element x in a ring R is called a clean element if there exists a unit u ∈ R and an idempotent e ∈ R such that x = e + u. A ring in which every element is a clean element is called a clean ring. Clean rings were introduced by Nicholson as examples of exchange rings [35]. Šter gives another characterization for clean elements in [45].
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Theorem 33 If R is a PI ring in which each element is the sum of two idempotents, then R/N(R) satisfies the identity x 3 = x where N(R) denotes the prime radical of R.
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Lemma 34 Let a be any element in a ring R. Then a is a clean element if and only if there exists an idempotent e ∈ R and a unit u ∈ R such that ua = eu + 1.
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Examples of Clean Rings.
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(1) The ring of linear transformations is a clean ring [38]. (2) Every unit regular ring is clean [6]. (3) If M is a module that is invariant under automorphisms of its injective envelope or pure-injective envelope then the endomorphism ring of M is a clean ring [17]. (4) If M is a flat module that is invariant under automorphisms of its cotorsion envelope then the endomorphism ring of M is a clean ring [17]. (5) The endomorphism ring of any continuous module is clean [7].
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We give below some basic facts about clean rings. Proposition 35 Let R be any ring.
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(1) R is clean if and only if R/J(R) is clean and idempotents lift modulo the Jacobson radical J(R) [35]. (2) If R is clean, then R is an exchange ring [35]. Bergman’s example mentioned in the first section is example of an exchange ring which is not clean. (3) If R is an exchange ring with central idempotents, then R is clean [35]. (4) R is semiperfect if and only if R is clean and R does not contain an infinite set of orthogonal idempotents [8].
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Properties of Clean Rings.
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(1) The center of a clean ring need not be clean [5]. (2) If R is a clean ring, then the matrix ring Mn (R) is also a clean ring [19]. (3) The corner ring of a clean ring need not be a clean ring [43].
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The most important recent development in the theory of clean rings has been the work of Šter who constructed for n ≥ 2, a ring R such that Mn (R) is clean but Mk (R) is not clean for k < n. This shows, in particular, that Theorem 36 (Šter, [44]) The property of being a clean ring is not a Morita-invariant property. The following question has been raised by Šter [44].
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Burgess and Rapahel [5] have shown that every ring can be embedded in a clean ring as an essential ring extension. Recall that a ring extension R ⊆ S is called an essential ring extension if for each nonzero ideal I of S, I ∩ R = 0. This shows the abundance of clean rings.
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Problem 37 Does there exist a ring R such that Mn (R) is clean for every n ≥ 2 but R is not clean?
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Problem 38 Characterize clean von Neumann regular rings.
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The following conjecture proposed in [42] is still open.
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There are numerous generalizations of clean rings available in the literature, but there are still some basic questions unanswered about clean rings. We list below some of them.
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Conjecture 39 Let R be a clean ring. Then the unit sum number of R is 2 if and only if R has no homomorphic image isomorphic to F2 . Very little is known about clean group rings. The following question is worth looking at. Problem 40 Characterize rings R and groups G such that the group ring R[G] is clean.
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Problem 41 Is T (X, F) a clean ring where F is a field?
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Let (X, d) be a locally finite metric space in the sense that all balls of finite radius are finite. Following Gromov [15], Ara et al. [2] defined the translation ring T (X, R) of X over R to be the ring of all square matrices [a(x, y)], indexed by X × X and with entries from R, such that a(x, y) = 0 whenever d(x, y) > l for some constant l depending on the matrix. The least such l is called the bandwidth of the matrix. Thus the Gromov translation ring T (X, R) is the ring of infinite matrices over R, indexed by X × X, with constant bandwidth. The translation ring also makes sense when d is just a locally finite pseudo metric. Ara et al. [2] showed that if X is a discrete tree and R is any von Neumann regular ring then the translation ring T (X, R) is an exchange ring. We would like to propose following questions.
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Problem 42 Let R be a ring with unit sum number 2. Is the translation ring T (X, R) also a ring with unit sum number 2?
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Problem 43 What is clean-dim(Mn (D)), where D is a division ring?
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3 Some Other Additive Representations
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3.1 Additive Regular Representation
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If R is a clean ring then a set of idempotents E ⊂ R such that each element in R can be expressed as e + u, where e ∈ E and u is a unit in R, will be called a cleangenerator set of R. If R is a clean ring, then the clean-dimension of R, denoted by clean-dim(R), is defined as clean-dim(R) = min{|E| : E is a clean-generator set of R}. It is easy to see that if D is a division ring, then clean-dim(D) = 2.
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Chatters, Ginn and Robson studied rings that are additively generated by their regular elements (see [9], and [37]). Recall that an element x in a ring R is called a regular element if x is not a left or right zero-divisor. Here are the main results concerning additive regular representation of elements in a ring. 419130_1_En_4_Chapter TYPESET
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Theorem 44 Let R be any ring.
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3.2 Additive Commutator Representation
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(1) If R is a prime right Goldie ring, then each element of R is the sum of at most two regular elements. (2) If R is a semiprime right Goldie ring, then R is generated additively by its regular elements if and only if R does not have a direct summand isomorphic to F2 ⊕ F2 . Furthermore, each element of R is the sum of two regular elements if R does not have a direct summand isomorphic to F2 . (3) If R is a left and right noetherian ring in which 2 is a regular element, then R is generated additively by its regular elements.
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Properties of commutator rings.
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(1) Any homomorphic image of a commutator ring is again a commutator ring. (2) Finite direct products of commutator rings are also commutator rings. (3) If R ⊆ S are rings such that R is a commutator ring and S is generated over R by elements centralizing R, then S is also a commutator ring. In particular, this means that matrix rings, group rings, and polynomial rings over commutator rings are also commutator rings. (4) Over any ring, the ring of infinite matrices that are both row-finite and columnfinite are commutator rings. (5) A finite-dimensional algebra over any field can never be a commutator ring. This implies, in particular, that no PI ring can be a commutator ring as every PI ring has a homomorphic image that is finite-dimensional over a field.
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Mesyan also proved that
Theorem 45 Any ring can be embedded in a commutator ring. The idea behind the proof of the above theorem is to construct for any ring R and any set I, a ring AI (R) = R < {xi }i∈I , {yi }i∈I : [xi , xj ] = [yi , yj ] = [xi , yj ] = 0 for i = j, and [xi , yi ] = 1 > which is a commutator ring. Let K be a field and E be an arbitrary directed graph. Let E 0 be the set of vertices, and E 1 be the set of edges of directed graph E. Consider two maps r : E 1 → E 0 and s : E 1 → E 0 . For any edge e in E 1 , s(e) is called the source of e and r(e) is called the range of e. If e is an edge starting from vertex v and pointing toward vertex w,
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In [33] a ring R is called a commutator ring if each element of R is a sum of additive commutators. One of the twelve open problems asked by Kaplansky in 1956 was whether there exists a division ring which is a commutator ring. Harris [22] answered the question of Kaplansky in the affirmative by constructing a division ring in which element is a sum of additive commutators. All the results in this subsection are due to Mesyan ([33] and [34]).
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then we imagine an edge starting from vertex w and pointing toward vertex v and call it the ghost edge of e and denote it by e∗ . We denote by (E 1 )∗ , the set of all ghost edges of directed graph E. If v ∈ E 0 does not emit any edges, i.e. s−1 (v) = ∅, then v is called a sink and if v emits an infinite number of edges, i.e. |s−1 (v)| = ∞, then v is called an infinite emitter. If a vertex v is neither a sink nor an infinite emitter, then v is called a regular vertex. The Leavitt path algebra of E with coefficients in K, denoted by LK (E), is the K-algebra generated by the sets E 0 , E 1 , and (E 1 )∗ , subject to the following conditions: vi vj = δij vi for all vi , vj ∈ E 0 . s(e)e = e = er(e) and r(e)e∗ = e∗ = e∗ s(e) for all e in E 1 . e∗i ej = δij r(ei ) for all ei , ej ∈ E 1 . If v ∈ E 0 is any regular vertex, then v = {e∈E 1 :s(e)=v} ee∗ .
Conditions If E 0 is finite, (CK1) and (CK2) are known as the0Cuntz-Krieger relations. then vi is an identity for LK (E) and if E is infinite, then E 0 generates a set of
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vi ∈E 0
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Theorem 46 The Leavitt path algebra LK (E) of a graph E over a field K is a commutator ring if and only if the following conditions hold; (1) (2) (3) (4)
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E is acyclic. E 0 contains only regular vertices. The characteristic p of K is not zero. For each u ∈ E 0 , there is an m ∈ N such that for all w ∈ E 0 satisfying d(u, w) = m + 1, the number of paths q = e1 e2 . . . ek ∈ P(E) such that s(q) = u, r(q) = w and s(e2 ), . . . , s(ek ) ∈ D(u, m) is a multiple of p.
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local units for LK (E). Let us denote the set of all paths in E by P(E). Given two vertices u, v ∈ E 0 such that there is a path p ∈ P(E) with s(p) = u and r(p) = v, let d(u, v) denote the length of the shortest such path. For each u ∈ E 0 and m ∈ N, denote D(u, m) = {v ∈ E 0 : d(u, v) ≤ m}. Mesyan studied commutator Leavitt path algebras in [34] and proved the following.
References
1. Ara, P., Goodearl, K.R., O’Meara, K.C., Pardo, E.: Diagonalization of matrices over regular rings. Linear Algbr. Appl. 265, 147–163 (1997) 2. Ara, P., Meara, K.C.O., Perera, F.: Gromov Translation algebras over discrete trees are exchange rings. Trans. Amer. Math. Soc. 356(5), 2067–2079 (2003) 3. Ashrafi, N., Vámos, P.: On the unit sum number of some rings. Quart. J. Math. 56, 1–12 (2005) 4. Belcher, P.: A test for integers being sums of direct units applied to cubic fields. Bull. Lond. Math. Soc. 6, 66–68 (1974) 5. Burgess, W., Raphael, R.: On embedding rings in clean rings. Comm. Algbr. 41(2), 552–564 (2013)
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6. Camillo, V.P., Khurana, D.: A characterization of unit regular rings. Comm. Algbr. 29(5), 2293–2295 (2001) 7. Camillo, V.P., Khurana, D., Lam, T.Y., Nicholson, W.K., Zhou, Y.: Continuous modules are clean. J. Algebr. 304, 94–111 (2006) 8. Camillo, V.P., Yu, H.P.: Exchange rings, units and idempotents. Comm. Algbr. 22(12), 4737– 4749 (1994) 9. Chatters, A.W., Ginn, S.M.: Rings generated by their regular elements. Glasgow Math. J. 25, 1–5 (1984) 10. Chen, H.: Decompositions of Countable Linear Transformations. Glasgow Math. J. 52, 427– 433 (2010) 11. Dieudonné, J.: La théorie de Galois des anneux simples et semi-simples. Comment. Math. Helv. 21, 154–184 (1948) 12. Fisher, J.W., Snider, R.L.: Rings generated by their units. J. Algebr. 42, 363–368 (1976) 13. Frei, C.: On rings of integers generated by their units. Bull. Lond. Math. Soc. 44(1), 167–182 (2012) 14. Goldsmith, B., Pabst, S., Scott, A.: Unit sum numbers of rings and modules, Quart. J. Math. Oxford, 49(2), 331–344(1998) 15. Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, vol. 2, Math. Soc. Lecture Notes Ser. 182, pp. 1–295, Cambridge University Press, London (1993), 16. Grover, H.K., Wang, Z., Khurana, D., Chen, J., Lam, T.Y.: Sums of units in rings. J. Algebr. Appl. 13, 10 (2014) 17. Guil Asensio, P.A., Keskin Tütüncü, D., Srivastava, A.K.: Modules invariant under automorphisms of their covers and envelopes. Israel J. Math. 206(1), 457–482 (2015) 18. Guil, P.A., Asensio, A.K.: Srivastava, Automorphism-invariant modules satisfy the exchange property. J. Algebr. 388, 101–106 (2013) 19. Han, J., Nicholson, W.K.: Extensions of clean rings. Comm. Algebr. 29(6), 2589–2595 (2001) 20. Hochschild, G.: Automorphisms of simple algebras. Trans. Amer. Math. Soc. 69, 292–301 (1950) 21. Handelman, D.: Perspectivity and cancellation in regular rings. J. Algebr. 48, 1–16 (1977) 22. Harris, B.: Commutators in division rings. Proc. Amer. Math. Soc. 9, 628–630 (1958) 23. Henriksen, M.: On a class of regular rings that are elementary divisor rings. Arch. Math. (Basel) 24, 133–141 (1973) 24. Henriksen, M.: Two classes of rings generated by their units. J. Algebra 31, 182–193 (1974) 25. Herwig, B., Ziegler, M.: A remark on sums of units. Arch. Math. 79(6), 430–431 (2002) 26. Hill, P.: Endomorphism ring generated by units. Trans. Amer. Math. Soc. 141, 99–105 (1969) 27. Hirano, Y., Tominaga, H.: Rings in which every element is the sum of two idempotents. Bull. Austral. Math. Soc. 37, 161–164 (1988) 28. Jarden, M., Narkiewicz, W.: On sums of units. Monatsh. Math. 150, 327–332 (2007) 29. Kaplansky, I.: Elementary divisors and modules. Trans. Amer. Math. Soc. 66, 464–491 (1949) 30. Khurana, D., Srivastava, A.K.: Right self-injective rings in which each element is sum of two units. J. Algebr. Appl. 6(2), 281–286 (2007) 31. Khurana, D., Srivastava, A.K.: Unit sum numbers of right self-injective rings. Bull. Austral. Math. Soc. 75(3), 355–360 (2007) 32. Khurana, S., Khurana, D., Nielsen, P. P.: Sums of units in self-injective rings. J. Algebr. Appl. 13 7 (2014) 33. Mesyan, Z.: Commutator rings. Bull. Austral. Math. Soc. 74, 279–288 (2006) 34. Mesyan, Z.: Commutator Leavitt path algebras. Algebr. Represent. Theory 16, 1207–1232 (2013) 35. Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229, 269– 278 (1977) 36. Raphael, R.: Rings which are generated by their units. J. Algebr. 28, 199–205 (1974) 37. Robson, J.C.: Sums of two regular elements. Glasgow Math. J. 25, 7–11 (1984) 38. Searcóid, M.: Perturbation of linear operators by idempotents. Irish Math. Soc. Bull., 39, 10–13 (1997)
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39. Siddique, F.: PhD. Dissertation, Saint Louis University, Saint Louis (2015) 40. Siddique, F., Srivastava, A. K.: Decomposing elements of a right self-injective ring. J. Algebr. Appl. 12(6) 10 (2013) 41. Skornyakov, L.A.: Complemented Modular Lattices and Regular Rings. Oliver & Boyd, Edinburgh (1964) 42. Srivastava, A.K.: A survey of rings generated by units. Annales de la Faculté des Sciences de Toulouse Mathématiques 19(S1), 203–213 (2010) 43. Šter, J.: Corner ring of a clean ring need not be clean. Comm. Algebr. 40(5), 1595–1604 (2012) 44. Šter, J.: The clean property is not a Morita invariant. J. Algebr. 420, 15–38 (2014) 45. Stringall, R.W.: Endomorphism rings of abelian groups generated by automorphism groups. Acta. Math. 18, 401–404 (1967) 46. Tang, G., Zhou, Y.: When is every linear transformation a sum of two commuting invertible ones? Linear Algebr. Appl. 439(11), 3615–3619 (2013) 47. Vámos, P.: 2-Good rings. Quart. J. Math. 56, 417–430 (2005) 48. Vámos, P., Wiegand, S.: Block diagonalization and 2-unit sums of matrices over prüfer domains. Trans. Amer. Math. Soc. 363, 4997–5020 (2011) 49. Wang, Z., Chen, J.L.: 2-Clean rings. Canad. Math. Bull. 52, 145–153 (2009) 50. Wang, L., Zhou, Y.: Decomposing linear transformations. Bull. Aust. Math. Soc. 83(2), 256– 261 (2011) 51. Wang, L., Zhou, Y.: Corrections: Decomposing linear transformations. Bull. Aust. Math. Soc. 85, 172–173 (2012) 52. Wolfson, K.G.: An ideal theoretic characterization of the ring of all linear transformations. Amer. J. Math. 75, 358–386 (1953) 53. Zelinsky, D.: Every linear transformation is sum of nonsingular ones. Proc. Amer. Math. Soc. 5, 627–630 (1954)
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