Algebra Defined by Ring Homomorphism is Associative

Theorem
Let $R$ be a commutative ring.

Let $\left({S, +, *}\right)$ 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 $\left({S_R, *}\right)$ be the algebra defined by the ring homomorphism $f$.

Then $\left({S_R, *}\right)$ is an associative algebra.

Proof
By definition, the multiplication of $\left({S_R, *}\right)$ is the ring product of $S$.

Thus it follows immediately from the fact that $S$ is a ring, that $\left({S_R, *}\right)$ 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