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)
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
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