(d) Prove Ker(f) = 1/J (e) Apply the Fundamental Homomorphism Theorem (FHT)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
Related questions
Question
Please solve d and e part
![Problem 3
Let R be any ring with ideals I and J such that J CI
Let 1/] = {a + J| a e 1}
Prove (R/J)/(I/J) = R/I as follows:
Define f: R/J→ R/I by f(a+J) = a +I
(a) Prove thatf is well-defined.
(b) Prove that f is a ring homomorphism.
(c) Prove that f is onto.
(d) Prove Ker(f) = 1/J
(e) Apply the Fundamental Homomorphism Theorem (FHT)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ca89b3e-0911-4c7e-b39f-9f10c4c7f9a5%2Fe71a3c0f-c736-4829-89a7-3a6518c1e558%2Frzoscn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3
Let R be any ring with ideals I and J such that J CI
Let 1/] = {a + J| a e 1}
Prove (R/J)/(I/J) = R/I as follows:
Define f: R/J→ R/I by f(a+J) = a +I
(a) Prove thatf is well-defined.
(b) Prove that f is a ring homomorphism.
(c) Prove that f is onto.
(d) Prove Ker(f) = 1/J
(e) Apply the Fundamental Homomorphism Theorem (FHT)
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