IGNOU BMTC-134 June 2022 Previous Year Question Paper

Q1.Which of the following statements are True or False ? Give reasons for your answer. 10x2=20

(a) Rhas a non-trivial subgroup of finite order.

(b) Every non-empty subset of Z has a least element. (०) Zinn = ZX Zp, AS groups, where m, ne N.

(d) IfGis a cyclic group, then it is isomorphic to each of its proper non-trivial subgroups.

(e) | Every ideal of the ring (R, +, +) is a normal subgroup of (R, +).

(f) If S is a subring of a commutative ring R, then S must be an ideal of R.

(g) IfR isa field and M is a maximal ideal of R, then (R/M) ~ R.

(h) IfRis an integral domain, then so is My(R).

(i) The set of integrable functions from R to R is a commutative ring w.r.t. pointwise addition and composition of functions. G@) ‘If (R, +, *) is a ring, then (R, +) and (R, -) are semigroups.

Q2.(a) Construct the Cayley table for the set F = {f,, fo, f3} w.r.t. the composition of maps, where the elements of F are functions from R\{0, 1} to R, defined as below : 1 Vv f(x) = —— V xe RN{O, I 1-5 Hence, check whether (F,-+) is a group or not. 8

(b) Forne N, is Z, a subring of Z? Give reasons for your answer.

Q3.(a) Let I be an ideal of a commutative ring R. Prove that vI ={xe R|x" € I for some n € N} is an ideal of R. Further, show that if R is a ring with unity and VI =R, thenI=R. 5

(b) Define a relation ‘~’ on M.(R) by “A~ B if and only if det

(A) = k det

(B) for some k € R*.” Check whether or not ‘~’ is an equivalence relation. If it is, find the equivalence class of 1 2 -1 -2 If ‘~ is not an equivalence relation, define another relation on M.(R) which is an equivalence relation.

Q4.(a) Let G be a group and let H be a normal subgroup of G. Show that if G/H is cyclic, then G need not be abelian. 3

(b) Let 5 o) 0, if ciseven n> Ba: 0 1, if cisodd where n 23. Show that is a group homomorphism. Also find a non-trivial element of ker 0. 4

(c) | Check whether or not 12 — 3x? — 9x4 + 25x5 is irreducible in QIx].

Q5.(a) Let G be a group and g € G. Use the Fundamental Theorem of Homomorphism to prove that Z 12 > is isomorphic to < g > if and only if g is of order 12. 7

(b) Is every prime ideal in a finite communicative ring with unity a maximal ideal ? Give reasons for your answer.

Q6.(a) Let D be an integral domain, and let K and L both be fields of fractions of D. Then prove that K = L. 3

(b) Consider the ring R=QIxl, and its ideal I = < x? —x >. Find an idempotent in R/ other than 0 or 1. 3

(c) Let G be a group and o¢€ AutG. Let Check whether or not H is a subgroup of G. If it is, then must H be normal in G ? If H is not a subgroup of G, give a proper normal subgroup of Aut Z.

Q7.(a) Find all possible group homomorphisms from Zg to Z45. 6

(b) Find all the proper ideals of Z/30Z. Further, which of these are maximal ideals and why?

Q8.(a) Let F bea field, F* = F\{0} and F’ = FX). Define ® on F by a @ b = a + b — ab. Show that (F’, ®) is a group, and that this group is isomorphic to (F’, +) 7

(b) Give an example, with justification, of a ring R which is not an integral domain and in which every ideal is a principal ideal.

Q9.(a) Let R=Z[i] and I=nR, wherene Z*. Show that a + ib € Lif and only if n|a and n|b. Further, show that R/I is a finite ring. 6

(b) Find Z(Dj9). Also find two distinct right cosets of Z(D 9) in Dy. 4 10

(a) Let Gbe an infinite group such that for any non-trivial subgroup N of G, |G: N| <«. Prove that if H < G, then H = {e} or H is infinite. Further, prove that if x is a non-trivial element of G, then O(x) is infinite. 5

(b) Use the Euclidean algorithm to find the g.c.d. ofx4+x+landx?+1lin QIxl. 3

(c) Give an example, with justification, of an element of S7 with order 12. 2 BMTC-134 6 BMTC-134 8 fax) = 2 vxe RX(0,0) 1-5 (@) “A~Bafe ok haa afe fet k eR* & far det

(A) = k det

(B)” RT Mo(R) R Fay ~ -1 -2 teat BMTC-134 10 l x2-x> fan Hifi | RIA O a