Editing Alternative algebra (section)

==Properties==
{{Redirect|Artin's theorem|Artin's theorem on primitive elements|Primitive element theorem}}
'''Artin's theorem''' states that in an alternative algebra the [[subalgebra]] generated by any two elements is [[associative algebra|associative]].<ref name=Sch29>Schafer (1995) p. 29</ref>  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 <math>x,y,z</math> in an alternative algebra associate (i.e., <math>[x,y,z] = 0</math>), 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.<ref name=Sch30>Schafer (1995) p. 30</ref>  The converse need not hold: the sedenions are power-associative but not alternative.

The [[Moufang identities]]
*<math>a(x(ay)) = (axa)y</math>
*<math>((xa)y)a = x(aya)</math>
*<math>(ax)(ya) = a(xy)a</math>
hold in any alternative algebra.<ref name=Sch28/>

In a unital alternative algebra, multiplicative [[inverse element|inverses]] are unique whenever they exist. Moreover, for any invertible element <math>x</math> and all <math>y</math> one has
:<math>y = x^{-1}(xy).</math>
This is equivalent to saying the associator <math>[x^{-1},x,y]</math> vanishes for all such <math>x</math> and <math>y</math>. 

If <math>x</math> and <math>y</math> are invertible then <math>xy</math> is also invertible with inverse <math>(xy)^{-1} = y^{-1}x^{-1}</math>. 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 (ring theory)|center]].<ref name=ZSSS151>Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p. 151</ref>
The structure theory of alternative rings is presented in the book ''Rings That Are Nearly Associative'' by Zhevlakov, Slin'ko, Shestakov, and Shirshov.<ref name=ZSSS>Zhevlakov, Slin'ko, Shestakov, Shirshov (1982)</ref>