T.Y. B.Sc. (Semester – IV) Examination, 2013 MATHEMATICS (Paper – IV) MT-344 : Ring Theory (2008 Pattern) (New Course) Time : 2 Hours
Max. Marks : 40 N.B. : 1) All questions are compulsory. 2) Figures to the right indicate full marks.
1. Attempt any five of the following :
10
a) Let a belong to a ring R. Let S = {x ∈ R | ax = 0} . Show that S is a subring of R. b) List all zero-divisors in 913. c) Find all maximal ideals in 912. d) Show that the function f :95 → 910 given by f(x) = 3x is not a homomorphism. e) Find zeros of x2 + 3x + 2 in 96. f) Determine whether the polynomial x4 + 3x + 3 is irreducible over Q. Justify. g) Show that in the ring 9[i]; 13 is reducible element. 2. Attempt any two of the following :
10
a) Prove that a finite integral domain is a field. b) Let R = {0, 2, 4, 6, 8} under addition and multiplication modulo 10. Write addition and multiplication table for R. Is R is a field ? Explain. c) Prove that the only ideals of a field F are {0} and F itself. P.T.O.
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