Flexible 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

Definition

Symbol-free definition

A flexible ring is a non-associative ring (i.e., a not necessarily associative ring) satisfying the following equivalent conditions:

1. Its multiplication gives a flexible magma.
2. The associator function is alternating in its first (i.e., leftmost) and third (i.e., rightmost) variable.

Definition with symbols

A flexible ring is a non-associative ring (i.e., a not necessarily associative ring) $R$ satisfying:

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

where $*$ is the multiplication in $R$.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Associative ring ring in the usual sense, with associativity
Alternative ring possibly nonassociative ring satisfying the alternative laws
Jordan ring ring that is multiplicatively a Jordan magma
Lie ring