Definition:Algebra Defined by Ring Homomorphism

Definition
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 module defined by the ring homomorphism $f$.

The algebra defined by $f$ is the algebra over $R$ defined as:
 * $(S_R, *)$

Also see

 * Algebra Defined by Ring Homomorphism is Algebra
 * Definition:Unital Associative Commutative Algebra

Properties

 * 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