The associator is a trilinear map given by By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal.
An alternating associator is always totally skew-symmetric.
The converse holds so long as the characteristic of the base field is not 2.
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.
[4] Conversely, any algebra for which this is true is clearly alternative.
It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra.
A generalization of Artin's theorem states that whenever three elements
), the subalgebra generated by those elements is associative.
A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.
[5] The converse need not hold: the sedenions are power-associative but not alternative.
The Moufang identities hold in any alternative algebra.
[2] In a unital alternative algebra, multiplicative inverses are unique whenever they exist.
The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop.
This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.
Kleinfeld's theorem states that any simple non-associative alternative ring is a generalized octonion algebra over its center.
[6] The structure theory of alternative rings is presented in the book Rings That Are Nearly Associative by Zhevlakov, Slin'ko, Shestakov, and Shirshov.
Then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1.