Equivalence of definitions of alternative ring
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
- is an Alternative magma (?), i.e., it satisfies the identities and for all .
- is both a Left-alternative magma (?) and a Flexible magma (?), i.e., it satisfies the identities and for all .
- is both a Right-alternative magma (?) and a Flexible magma (?), i.e., it satisfies the identities and for all .
- The associator function on is an alternating function on any two of its variables.
The associator function:
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||alternating in first and second variable|
|right-alternative||alternating in second and third variable|
|flexible||alternating in first and third variable|
Fact (1) now tells us that any two of these conditions must imply the third, completing the proof.