Algebra Defined by Ring Homomorphism is Associative

Theorem
Let $R$ be a commutative ring.

Let $(S, +, *)$ be a ring with unity.

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

Let the image of $f$ be a subset of the center of $S$.

Let $(S_R, *)$ be the algebra defined by the ring homomorphism $f$.

Then $(S_R, *)$ is an associative algebra.

Proof
By definition, the multiplication of $(S_R, *)$ is the multiplication of $S$.

Thus it follows immediately from the fact that $S$ is a ring, that $(S_R, *)$ is an associative algebra.

Also see

 * Algebra Defined by Ring Homomorphism is Algebra
 * Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary
 * Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative