Artin's theorem on alternative rings
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
The following are equivalent for a non-associative ring (i.e., a not necessarily associative ring ):
- is an alternative ring, i.e., it is an Alternative magma (?) under its multiplication operation.
- is a diassociative ring, i.e., the subring generated by any two elements is associative. Equivalently, is a Diassociative magma (?) under the multiplication operation.
- Artin's generalized theorem on alternative rings: This says that if are (possibly equal, possibly distinct) elements of an alternative ring such that where denotes the associator, then the subring generated by is associative.
Given: An alternative ring . Elements (possibly equal, possibly distinct)
To prove: The subring of generated by and is associative
Proof: Set and use Fact (1), along with the observation that, because is alternative, we must have .