Definition:Algebra over Ring

Definition
Let $R$ be a commutative ring.

An algebra over $R$ is an ordered pair $\left({A, *}\right)$ where:


 * $A$ is an $R$-module
 * $*: A^2 \to A$ is an $R$-bilinear mapping

Commutativity of the Ring
Because the definition of bilinear mapping is intricate in the case of noncommutative rings, it is important that $R$ be commutative. It does not necessarily have to be a ring with unity.

Also defined as
Especially in commutative algebra, an algebra over a commutative ring with unity $R$ is often defined as a ordered pair $(S, f)$ where $S$ is a commutative ring with unity and $f : R \to S$ is a unital ring homomorphism.

Also see

 * Definition:Algebra over Field
 * Definition:Unitary Algebra

It can be considered to be a generalization of an algebra over a field in which:
 * the vector space is replaced by a module
 * the field is replaced by a commutative ring.