Left alternative ring
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
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
|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|