Let R and S be rings, F a field, and f : R → S, g : F → R ring homomorphims. (a) Prove that, for any r ∈ R and n ∈ Z, f (nr) = nf (r). (b) Prove that for c ∈ F with c ̸= 0, if f (c) = 0R, then f is the zero map, i.e. for all x ∈ F , f (x) = 0R. (c) Prove that f if f is not the zero homomorphism, then f is injective.

Let R and S be rings, F a field, and f : R → S, g : F → R ring homomorphims. (a) Prove that, for any r ∈ R and n ∈ Z, f (nr) = nf (r). (b) Prove that for c ∈ F with c ̸= 0, if f (c) = 0R, then f is the zero map, i.e. for all x ∈ F , f (x) = 0R. (c) Prove that f if f is not the zero homomorphism, then f is injective.

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 28E

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