Let f : R → S be a surjective ring homomorphism and suppose I ⊆ R and J ⊆ S are ideals. (a) Prove that the image of I, f [I] = { s ∈ S : ∃r ∈ R, f (r) = s }, is an ideal of S. (b) Prove also that f -1[J] = { r ∈ R : f (r) ∈ J } is an ideal of R. Note that when J = (0S), then f -1[(0S )] is the kernel of f, ker(f ).

Let f : R → S be a surjective ring homomorphism and suppose I ⊆ R and J ⊆ S are ideals. (a) Prove that the image of I, f [I] = { s ∈ S : ∃r ∈ R, f (r) = s }, is an ideal of S. (b) Prove also that f -1[J] = { r ∈ R : f (r) ∈ J } is an ideal of R. Note that when J = (0S), then f -1[(0S )] is the kernel of f, ker(f ).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 32E: 32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing...

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