Theorem 15.1 Properties of Ring Homomorphisms Let o be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S. 1. For any r E R and any positive integer n, þ(nr) = nd(r) and 6(r") = (6(r))". 2. (A) = {d(a) | a E A} is a subring of S. 3. If A is an ideal and o is onto S, then 4(A) is an ideal. 4. 4-(B) = {r ER| 4(r) E B} is an ideal of R. 5. If R is commutative, then 4(R) is commutative. 6. If R has a unity 1, S + {0}, and is onto, then 4(1) is the unity of S. 7. o is an isomorphism if and only if o is onto and Ker = {r ER|¢(r) = 0} = {0}. 8. If o is an isomorphism from R onto S, then -1 is an isomorphism from S onto R.

Theorem 15.1 Properties of Ring Homomorphisms Let o be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S. 1. For any r E R and any positive integer n, þ(nr) = nd(r) and 6(r") = (6(r))". 2. (A) = {d(a) | a E A} is a subring of S. 3. If A is an ideal and o is onto S, then 4(A) is an ideal. 4. 4-(B) = {r ER| 4(r) E B} is an ideal of R. 5. If R is commutative, then 4(R) is commutative. 6. If R has a unity 1, S + {0}, and is onto, then 4(1) is the unity of S. 7. o is an isomorphism if and only if o is onto and Ker = {r ER|¢(r) = 0} = {0}. 8. If o is an isomorphism from R onto S, then -1 is an isomorphism from S onto R.

Name: Prove this theorem. Theorem 15.1 Properties of Ring HomomorphismsLet o
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Description: Prove this theorem. Theorem 15.1 Properties of Ring HomomorphismsLet o be a ring homomorphism from a ring R to a ring S. Let A be asubring of R and let B be an ideal of S.1. For any r E R and any positive integer n, þ(nr) = nd(r) and6(r") = (6(r))".2

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 30E: a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R....

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