Algebra Defined by Ring Homomorphism is Algebra

Theorem
Let $R$ be a commutative ring.

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

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 algebra over $R$.

Also see

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