# Left 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

### Symbol-free definition

A non-associative ring (i.e., a not necessarily associative ring) is termed a **left-alternative ring** if it satisfies the following equivalent conditions:

- The associator is an alternating function of its first two variables.
- Its multiplicative magma is a left-alternative magma.

### Definition with symbols

A non-associative ring (i.e., a *not necessarily associative* ring ) is termed a **left-alternative ring** if it satisfies the following identity:

Note that are allowed to be equal. Here, is the multiplication of .

## Relation with other properties

### Stronger properties

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

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

Alternative ring | both left-alternative and right-alternative | |FULL LIST, MORE INFO |