Problem 3 Let R be any ring with ideals I and J such that JcI Let 1/J = {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 that f 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)
Problem 3 Let R be any ring with ideals I and J such that JcI Let 1/J = {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 that f 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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 3
Let R be any ring with ideals I and J such that JcI
Let 1/J = {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 that f is well-defined.
(b) Prove thatf 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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