# Category:Algebras

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This category contains results about **algebras over rings and fields** in the context of **Abstract Algebra**.

Definitions specific to this category can be found in Definitions/Algebras.

Let $R$ be a commutative ring.

An **algebra over $R$** is an ordered pair $\struct {A, *}$ where:

- $A$ is an $R$-module
- $*: A^2 \to A$ is an $R$-bilinear mapping

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Algebras"

The following 23 pages are in this category, out of 23 total.

### A

- Algebra Defined by Ring Homomorphism is Algebra
- Algebra Defined by Ring Homomorphism is Associative
- Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative
- Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary
- Artin's Theorem on Alternative Algebras
- Associative Algebra has Multiplicative Inverses iff Unitary Division Algebra