# Category:Algebras

From ProofWiki

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 $\left({A, *}\right)$ where:

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

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Algebras"

The following 22 pages are in this category, out of 22 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

### C

### N

- Nicely Normed Alternative Algebra is Normed Division Algebra
- Nonassociative Division Algebra with Real Scalars has Dimension of Power of 2
- Nonassociative Division Algebra with Real Scalars has Dimension of Power of 2/Historical Note
- Norm of Unit of Normed Division Algebra
- Normed Division Algebra is Division Algebra