# 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 unital associative commutative algebra.

## Also see

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.
• Results about algebras can be found here.