# Definition:Commutative Algebra (Abstract Algebra)

This page is about Commutative Algebra in the context of Abstract Algebra. For other uses, see Commutative Algebra.

## Definition

Let $R$ be a commutative ring.

Let $\struct {A_R, \oplus}$ be an algebra over $R$.

Then $\struct {A_R, \oplus}$ is a commutative algebra if and only if $\oplus$ is a commutative operation.

That is:

$\forall a, b \in A_R: a \oplus b = b \oplus a$