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 $R[X]$ and $S[X]$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$.

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

is a ring homomorphism.

Proof
Let $f = a_0 + \cdots + a_n X^n$, $g = b_0 + \cdots + b_mX^m \in R[X]$.

We have:

and

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