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Problem 11

Avg difficulty 3.0

Let RR and SS be commutative rings, and let ϕ:R→S\phi:R\to S be a ring homomorphism. Define the map ψ:R[x]→S[x]\psi:R[x]\to S[x] where

ψ(∑i=0naixi)=∑i=0nϕ(ai)xi.\psi\left(\sum_{i=0}^{n}a_{i}x^{i}\right)=\sum_{i=0}^{n}\phi(a_{i})x^{i}.

  1. Show that ψ\psi is a ring homomorphism.

  2. If ϕ\phi is an isomorphism, prove that R[x]≅S[x]R[x]\cong S[x].

Kan, A Complete Introduction to Abstract Algebra, Rings, Polynomial Rings, Problem 11

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.