Left alternative ring

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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

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:

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

Definition with symbols

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

\! (x * x) * y = x * (x * y) \ \forall \ x,y \in R

Note that x,y are allowed to be equal. Here, * is the multiplication of R.

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