BSc Mathematics SEM VI 2018 19 May 2018-19 MATHEMATICS ALGEBRA Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks for respective subquestions

1. Q1 Choose the correct option. Attempt all the subquestions
   • (i) Let H be a normal subgroup of G. Let o(aH) =3 in $ and |H| = 10, then order of a is 2
   • (c) one of 3, 6, 15 or 30
   • (d) None of these
   • (ii) Which of the following is not true for a normal subgroup H ofa group 2
   • (a) C for each a G
   • (b) = HA for each a G
   • (c) Every left coset of H in G is also a right coset of H in G aH = Ha for each G
   • (d) G/H is Abelian
   • (iii) Which of the following is not true? 2
   • (a) is isomorphic to A3
   • (b) is isomorphic to < (2 13 >, a subgroup of
   • (c) V4 is isomorphic to (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} a sub
   • (d) is isomorphic to a subgroup of
   • (iv) The group of units of a ring is 2
   • (a) Abelian but may not be cyclic
   • (b) Cyclic
   • (c) may not be Abelian
   • (d) finite
   • (v) Consider the ideals of ring of integers J = 6Z and J = 10Z, then 2
   • (d) None of these
   • (vi) In the polynomial ring consider J = : f(0) then 2
   • (a) I is an ideal
   • (b) J is a maximal ideal
   • (c) I is ideal but neither prime ideal nor maximal
   • (d) J is prime ideal but not maximal ideal
   • (vii) Which of the following is true in 2
   • (a) is irreducible but not prime
   • (b) 2+ is prime
   • (c) 3 is prime
   • (d) 2+ /—5 is reducible
   • (viii) The number of maximal ideals in R x R is 2
   • (ix) The field of quotients of is 2
   • (d) None of these
   • (x) Let (2? in 1 <n Then, is a field if 2
   • (b) for alln <5
2. Q2 (a) Answer any ONE
   • (i) Let G and G’ be groups and f : G > G’ be an onto homomorphism. Prove that if H is a subgroup G then f(H) = {f(h): he H}isa subgroup of G’ and f(Ha) = f(H)f(a) for each a G. Further, if H is normal in G then f(#) is normal in G’. Give example to show that f(H) need not be normal in G’ if f is not onto 8
   • (ii) If a such that o(a) = m,o(b) = n, then prove that (a,b)* = for every k N and o(a,b) = lem(m,n). Hence prove that, are cyclic then x is cyclic if and only if and are relatively prime 8
   • (b) Answer any TWO
   • (i) Show that kernel of a group homomorphism G’ is a normal subgroup of G. Also show that for any normal subgroup G there is a group homomrphism 7 : G — G/H such that ker = H 6
   • (ii) If G/Z(G) is cyclic then prove that G is Abelian. 6
   • (iii) Show that order of each element of the quotient group is finite
   • (iv) Show that {e,b} is normal in a?} but not normal in fe,a, b, ab, a*b, where at =e = aba = b 6
3. Q3 (a) Answer any ONE
   • (i) Define characteristic of a ring R. Show that, characteristic of a ring R is n if and only if the order of the multiplicative identity of R is n in the group (R,+). Give example of an infinite ring with 8
   • (ii) Let R be a commutative ring. If are ideals in R, Show that In and IJ are ideals of R, where Further if R= J+ J, show that 1Q J = IJ 8
   • (b) Answer any TWO
   • (i) Let A be a subring and B be an ideal of a ring R. Then prove that AN B is an ideal of A/(AN B) ~ (A+ B)/B 6
   • (ii) Let R,R’ be commutative rings and f : R > R’ be a ring homo- morphism. Show that 6
   • (1) If f is surjective, J is an ideal of R, then is an ideal of R’ (II) If J’ is an ideal of then f~'(J’) is an ideal of R
   • (iii) Let R = Z[V2| = {a + bV2: a,b Z} and I = Z,ais even}. Show that the quotient ring R/I is isomorphic to Zy 6
   • (iv) Let R be a commutative ring with prime characteristic p and R be defined as f(a) = a? for a R. Show that f is a ring 6
4. Q4 (a) Answer any ONE
   • (i) Show that an ideal P in a commutative ring R is a prime ideal if and only if the quotient ring R/P is integral domain Further prove that in a finite commutative ring every prime ideal is maximal 8
   • (ii) Define irreducible polynomial. Let be a field. Show that < > is a field if and only if f(x) is irreducible over F’ 8
   • (b) Answer any TWO
   • (i) Let R,S be commutative rings. And f be an onto ring homomorphism. Prove that, if M is a maximal ideal in S then, is a maximal ideal in R 6
   • (ii) Show that the only irreducible polynomials in are a linear poly- nomial or quadratic polynomial such that b?—4c < 0, 6
   • (iii) Show that in Z/i], 3 is irreducible but 2 is not irreducible. 6
   • (iv) Show that < x,2 >, the ideal generated by and 2 is a maximal ideal of Z[x]. Further show that this ideal is not principal ideal 6
5. Q5 Answer any FOUR
   • (a) Let G be a group anf H be a normal subgroup of G. Then prove that 5
   • (p) (Ha)” = Ha” for all n Z
   • (q) o(Ha) divides o(a)
   • (b) Find a subgroup of order 9 in x x 5
   • (c) Show that a finite field of size 8 has characteristic 2. 5
   • (d) Determine all the ideals of + 32? — 4) by stating the results 5
   • (e) Let R be commutative and be ideal of R and P is a prime ideal of Rthat contains J. Prove that either J C P or J C P 5
   • (f) Let F bea field. Show that every ideal of F[z] is a principal ideal. 5

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