Induced Homomorphism of Polynomial Forms

Theorem
Let $R$ and $S$ be commutative rings with unity.

Let $\phi: R \to S$ be a ring homomorphism.

Let $R \sqbrk X$ and $S \sqbrk X$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$.

Then the map $\overline \phi: R \sqbrk X \to S \sqbrk X$ given by:
 * $\map {\overline \phi} {a_0 + a_1 X + \cdots + a_n X^n} = \map \phi {a_0} + \map \phi {a_1} X + \cdots + \map \phi {a_n} X^n$

is a ring homomorphism.

Also see

 * Definition:Polynomial Evaluation Homomorphism