Abstract Algebra: Theory and Applications · Problem 18

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · ring-homomorphisms

Problem

Let ϕ:R→S\phi : R \to S be a ring homomorphism. Prove each of the following statements. (a) If RR is a commutative ring, then ϕ(R)\phi(R) is a commutative ring. (b) ϕ(0)=0\phi(0) = 0. (c) Let 1R1_R and 1S1_S be the identities for RR and SS, respectively. If ϕ\phi is onto, then ϕ(1R)=1S\phi(1_R) = 1_S. (d) If RR is a field and ϕ(R)≠0\phi(R) \neq 0, then ϕ(R)\phi(R) is a field. Hint. (a) ϕ(a)ϕ(b)=ϕ(ab)=ϕ(ba)=ϕ(b)ϕ(a)\phi(a)\phi(b) = \phi(ab) = \phi(ba) = \phi(b)\phi(a).

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