3.4 Homomorphisms

1.
a. No. θ is a general ring homomorphism, because 42 = 4 in img But θ(1) = 4, and 4 ≠ 1 in img
c. No. θ[(r, s) · (r′, s′)] = rr′ + ss′ need not equal

img

e. Yes. θ(fg) = (fg)(1) = f(1)g(1) = θ(f) · θ(g). Similarly for f ± g. The unity of img is img given by img. Thus img
2.
a. Write θ(1) = e, and let s img S. Then s = θ(r) for some r img R (θ is onto) so

img

Hence e is the unity of S.
3. If img is a general ...

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