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 thatf is onto. (d) Prove Ker(f) = 1/J (e) Apply the Fundamental Homomorphism Theorem (FHT)
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 thatf is onto. (d) Prove Ker(f) = 1/J (e) Apply the Fundamental Homomorphism Theorem (FHT)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![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%2F7c6e6fb6-6b02-46fa-b16a-f866dafd61de%2Fxnpnwod_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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