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 $R \left[{X}\right]$ and $S \left[{X}\right]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$.

Then the map $\overline{\phi}: R \left[{X}\right] \to S \left[{X}\right]$ given by:
 * $\overline{\phi} \left({a_0 + a_1 X + \cdots + a_n X^n}\right) = \phi \left({a_0}\right) + \phi \left({a_1}\right) X + \cdots + \phi \left({a_n}\right) X^n$

is a ring homomorphism.

Proof
Let $f = a_0 + \cdots + a_n X^n$, $g = b_0 + \cdots + b_m X^m \in R \left[{X}\right]$.

We have:

and

Thus $\overline{\phi}$ is a ring homomorphism.