Definition:Algebra Defined by Ring Homomorphism

Definition

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 $S_R$ be the module defined by the ring homomorphism $f$.

The algebra defined by $f$ is the algebra over $R$ defined as:

$\struct {S_R, *}$