# Artin's theorem on alternative rings

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term alternative ring

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a non-associative ring (i.e., a not necessarily associative ring ):

- is an alternative ring, i.e., it is an Alternative magma (?) under its multiplication operation.
- is a diassociative ring, i.e., the subring generated by any two elements is associative. Equivalently, is a Diassociative magma (?) under the multiplication operation.

## Facts used

- Artin's generalized theorem on alternative rings: This says that if are (possibly equal, possibly distinct) elements of an alternative ring such that where denotes the associator, then the subring generated by is associative.

## Proof

**Given**: An alternative ring . Elements (possibly equal, possibly distinct)

**To prove**: The subring of generated by and is associative

**Proof**: Set and use Fact (1), along with the observation that, because is alternative, we must have .