Artin's theorem on alternative rings

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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 are equivalent for a non-associative ring R (i.e., a not necessarily associative ring R):

  • R is an alternative ring, i.e., it is an Alternative magma (?) under its multiplication operation.
  • R is a diassociative ring, i.e., the subring generated by any two elements is associative. Equivalently, R is a Diassociative magma (?) under the multiplication operation.

Facts used

  1. Artin's generalized theorem on alternative rings: This says that if x,y,z are (possibly equal, possibly distinct) elements of an alternative ring such that a(x,y,z) = 0 where a denotes the associator, then the subring generated by x,y,z is associative.

Proof

Given: An alternative ring R. Elements x,y \in R (possibly equal, possibly distinct)

To prove: The subring of R generated by x and y is associative

Proof: Set z = y and use Fact (1), along with the observation that, because R is alternative, we must have a(x,y,y) = 0.