Algebra Defined by Ring Homomorphism is Algebra

Theorem
Let $R$ be a commutative ring.

Let $\struct {S, +, *}$ 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 $\struct {S_R, *}$ be the algebra defined by the ring homomorphism $f$.

Then $\struct {S_R, *}$ 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