# Alternative ring

From Groupprops

This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.

View other non-associative ring properties

## Contents

## Definition

An **alternative ring** is a non-associative ring (i.e., a not necessarily associative ring) that, under its multiplication , satisfies one of the following equivalent conditions:

- 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.
- The subring of generated by any two elements of is an associative ring.

### Equivalence of definitions

- The equivalence of definitions (1) -- (4) is proved by showing that they are all equivalent to (4).
`Further information: equivalence of definitions of alternative ring` - The equivalence with definition (5) is Artin's theorem on alternative rings.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Associative ring | associativity holds universally | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Left-alternative ring | ||||

Right-alternative ring | ||||

Flexible ring | ||||

Power-associative ring |