# Equivalence of definitions of alternative ring

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 (all definitions of alternative ring) are equivalent for a non-associative ring with multiplication denoted by :

- is an Alternative magma (?), i.e., it satisfies the identities and for all .
- is both a Left-alternative magma (?) and a Flexible magma (?), i.e., it satisfies the identities and for all .
- is both a Right-alternative magma (?) and a Flexible magma (?), i.e., it satisfies the identities and for all .
- The associator function on is an alternating function on any two of its variables.

## Facts used

## Proof

The associator function:

is additive separately in each variable, i.e., it is multi-additive. Here is what each condition says:

Condition name | Equational form | What does it tell us about the associator function? |
---|---|---|

left-alternative | alternating in first and second variable | |

right-alternative | alternating in second and third variable | |

flexible | alternating in first and third variable |

Fact (1) now tells us that any two of these conditions must imply the third, completing the proof.