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

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

  1. (R,*) is an alternative magma, i.e., it satisfies the identities x * (x * y) = (x * x) * y and x * (y * y) = (x * y) * y for all x,y \in R.
  2. (R,*) is both a left-alternative magma and a flexible magma, i.e., it satisfies the identities x * (x * y) = (x * x) * y and x * (y * x) = (x * y) * x for all x,y \in R.
  3. (R,*) is both a right-alternative magma and a flexible magma, i.e., it satisfies the identities x * (y * y) = (x * y) * y and x * (y * x) = (x * y) * x for all x,y \in R.
  4. The associator function on R is an alternating function on any two of its variables.
  5. The subring of R generated by any two elements of R is an associative ring.

Equivalence of definitions

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