Introduction to Abstract Algebra, Solutions Manual, 4th Edition

3.4 Homomorphisms

1.

a. No. θ is a general ring homomorphism, because 4² = 4 in But θ(1) = 4, and 4 ≠ 1 in

c. No. θ[(r, s) · (r′, s′)] = rr′ + ss′ need not equal

e. Yes. θ(fg) = (fg)(1) = f(1)g(1) = θ(f) · θ(g). Similarly for f ± g. The unity of is given by . Thus

2.

a. Write θ(1) = e, and let s S. Then s = θ(r) for some r R (θ is onto) so

Hence e is the unity of S.

3. If is a general ...