a) Let R and S be any rings. Show that the zero map, (ie, f: RS defined by f(r) = 0sfor all rER) is a ring homomorphism.b) Show Z → M (Z) defined by f(x) = [02] is a ring homomorphism.c) Prove that if f: RS is any ring homomorphism, f(0R) = 0s.d) Let f: Z16 Z4 x Z4 be a ring homomorphism. By (c), f(0)=(0,0). Iff(1) = (1,1), find the image of the other elements of Z16. Is f injective? Is fsurjective? Justify your answers.

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Answered: a) Let R and S be any rings. Show that…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 10E: Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove...

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Question

a) Let R and S be any rings. Show that the zero map, (ie, f: RS defined by f(r) = 0s
for all rER) is a ring homomorphism.
b) Show Z → M (Z) defined by f(x) = [02] is a ring homomorphism.
c) Prove that if f: RS is any ring homomorphism, f(0R) = 0s.
d) Let f: Z16 Z4 x Z4 be a ring homomorphism. By (c), f(0)=(0,0). If
f(1) = (1,1), find the image of the other elements of Z16. Is f injective? Is f
surjective? Justify your answers.

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