Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary

Theorem

Let $R$ be a commutative ring.

Let $\struct {S, +, *}$ be a ring with unity.

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

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

Then $\struct {S_R, *}$ is a unitary algebra.

Proof

By definition, the multiplication of $\struct {S_R, *}$ is the ring product of $S$.

Thus it follows immediately from the fact that $S$ is a ring with unity, that $\struct {S_R, *}$ is a unitary algebra.

$\blacksquare$