Artin's theorem on alternative rings: Difference between revisions

Latest revision as of 19:42, 15 May 2012

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

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 $a (x, y, y) = 0$ .

@@ Line 7: / Line 7: @@
 * <math>R</math> is an [[alternative ring]], i.e., it is an [[fact about::alternative magma]] under its multiplication operation.
 * <math>R</math> is a diassociative ring, i.e., the subring generated by any two elements is associative. Equivalently, <math>R</math> is a [[fact about::diassociative magma]] under the multiplication operation.
+==Facts used==
+# [[uses::Artin's generalized theorem on alternative rings]]: This says that if <math>x,y,z</math> are (possibly equal, possibly distinct) elements of an [[alternative ring]] such that <math>a(x,y,z) = 0</math> where <math>a</math> denotes the [[associator]], then the subring generated by <math>x,y,z</math> is associative.
+==Proof==
+'''Given''': An alternative ring <math>R</math>. Elements <math>x,y \in R</math> (possibly equal, possibly distinct)
+'''To prove''': The subring of <math>R</math> generated by <math>x</math> and <math>y</math> is associative
+'''Proof''': Set <math>z = y</math> and use Fact (1), along with the observation that, because <math>R</math> is alternative, we must have <math>a(x,y,y) = 0</math>.