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Exercise 13.1. (Chinese Remainder Theorem.) Let R be a ring. Given two ideals I and J of R, suppose I+J = R. Show that the factor ring R/In J is isomorphic to the product ring (R/I) × (R/J). (Hint: Define : R → (R/I) × (R/J) by o(a) := (a + I, a + J) for every a E R. Show that is a surjective ring homomorphism with ker() = In J, and then by the fundamental homomorphism theorem.)

Exercise 13.1. (Chinese Remainder Theorem.) Let R be a ring. Given two ideals I and J of R, suppose I+J = R. Show that the factor ring R/In J is isomorphic to the product ring (R/I) × (R/J). (Hint: Define : R → (R/I) × (R/J) by o(a) := (a + I, a + J) for every a E R. Show that is a surjective ring homomorphism with ker() = In J, and then by the fundamental homomorphism theorem.)

Elements Of Modern Algebra
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 4E: Exercises If and are two ideals of the ring , prove that is an ideal of .
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Exercise 13.1. (Chinese Remainder Theorem.) Let R be a ring. Given two ideals I and J of R, suppose I+J = R. Show that the factor ring R/In J is isomorphic to the product ring (R/I) × (R/J). (Hint: Define : R → (R/I) × (R/J) by o(a) := (a + I, a + J) for every a E R. Show that is a surjective ring homomorphism with ker() = In J, and then by the fundamental homomorphism theorem.)
Transcribed Image Text:Exercise 13.1. (Chinese Remainder Theorem.) Let R be a ring. Given two ideals I and J of R, suppose I+J = R. Show that the factor ring R/In J is isomorphic to the product ring (R/I) × (R/J). (Hint: Define : R → (R/I) × (R/J) by o(a) := (a + I, a + J) for every a E R. Show that is a surjective ring homomorphism with ker() = In J, and then by the fundamental homomorphism theorem.)
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