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:

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

Also see

• Definition:Associative Algebra

• Results about commutative algebras can be found here.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem