Equivalence of definitions of alternative ring

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This article gives a proof/explanation of the equivalence of multiple definitions for the term alternative ring
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following (all definitions of alternative ring) are equivalent for a non-associative ring R with multiplication denoted by *:

  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.

Facts used

  1. Alternating function condition is transitive

Proof

The associator function:

a(x,y,z) = ((x * y) * z) - (x * (y * z))

is additive separately in each variable, i.e., it is multi-additive. Here is what each condition says:

Condition name Equational form What does it tell us about the associator function?
left-alternative (x * x) * y = x * (x * y) alternating in first and second variable
right-alternative (x * y) * y = x * (y * y) alternating in second and third variable
flexible x * (y * x) = (x * y) * x alternating in first and third variable

Fact (1) now tells us that any two of these conditions must imply the third, completing the proof.