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
Sources who only deal with rings with unity often define an algebra as one whose underlying module is unital.

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. At, this is called a unital associative commutative algebra.

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.