Definition:Associative Algebra

This page is about associative algebra. For other uses, see associative.

Definition

Let $R$ be a commutative ring.

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

Then $\struct {A_R, *}$ is an associative algebra if and only if $*$ is an associative operation.

That is:

$\forall a, b, c \in A_R: \paren {a * b} * c = a * \paren {b * c}$

Also see

• Definition:Commutative Algebra (Abstract Algebra)
• Definition:Commutative Algebra (Mathematical Branch)

• Results about associative algebras can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): associative: 2.
• 2002: John C. Baez: The Octonions: 1.1 Preliminaries
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): algebra: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): associative algebra
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations