Answer to Question #47341 in Abstract Algebra for ayush kumar

Question

(a) Let R and R' be commutative rings and f : R --> R' be a ring homomorphism. Prove or disprove that if I is an ideal of R, then f(I) is an ideal of R'.

(b) Let G be a group and Z (G) ={a E G Ixa=ax for all x E G}. If G/Z (G) is cyclic, then show that G is an Abelian group.

(c) Iet (D. 6) be a Euclidean domain. Prove that for every integer n such that delta (1)+n greater than equal to 0, the function Fn:D\{0}->Z:Fn (a)=6(delta)a+n is a Euclidean valuation on D.

(d) Let G he a cyclic group of order 6. Can G be isomorphic to a subgroup of S7 ? lustily your answer.