Alternative ring: Difference between revisions

Latest revision as of 01:45, 18 February 2011

This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties

Definition

An alternative ring is a non-associative ring $R$ (i.e., a not necessarily associative ring) that, under its multiplication $*$ , satisfies one of the following equivalent conditions:

$(R, *)$ is an alternative magma, i.e., it satisfies the identities $x * (x * y) = (x * x) * y$ and $x * (y * y) = (x * y) * y$ for all $x, y \in R$ .
$(R, *)$ is both a left-alternative magma and a flexible magma, i.e., it satisfies the identities $x * (x * y) = (x * x) * y$ and $x * (y * x) = (x * y) * x$ for all $x, y \in R$ .
$(R, *)$ is both a right-alternative magma and a flexible magma, i.e., it satisfies the identities $x * (y * y) = (x * y) * y$ and $x * (y * x) = (x * y) * x$ for all $x, y \in R$ .
The associator function on $R$ is an alternating function on any two of its variables.
The subring of $R$ generated by any two elements of $R$ is an associative ring.

Equivalence of definitions

The equivalence of definitions (1) -- (4) is proved by showing that they are all equivalent to (4). Further information: equivalence of definitions of alternative ring
The equivalence with definition (5) is Artin's theorem on alternative rings.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Associative ring	associativity holds universally			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Left-alternative ring
Right-alternative ring
Flexible ring
Power-associative ring

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+{{non-associative ring property}}
 ==Definition==
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 # <math>(R,*)</math> is both a [[defining ingredient::left-alternative magma]] and a [[defining ingredient::flexible magma]], i.e., it satisfies the identities <math>x * (x * y) = (x * x) * y</math> and <math>x * (y * x) = (x * y) * x</math> for all <math>x,y \in R</math>.
 # <math>(R,*)</math> is both a [[defining ingredient::right-alternative magma]] and a [[defining ingredient::flexible magma]], i.e., it satisfies the identities <math>x * (y * y) = (x * y) * y</math> and <math>x * (y * x) = (x * y) * x</math> for all <math>x,y \in R</math>.
+# The [[defining ingredient::associator on a non-associative ring|associator function]] on <math>R</math> is an alternating function on any two of its variables.
 # The subring of <math>R</math> generated by any two elements of <math>R</math> is an [[associative ring]].
 ===Equivalence of definitions===
-* The equivalence of definitions (1) -- (3) follows by looking at the [[associator on a non-associative ring]] and proving that any of the conditions is equivalent to that associator being an alternating function of its inputs. {{further|[[equivalence of definitions of alternative ring]]}}
+* The equivalence of definitions (1) -- (4) is proved by showing that they are all equivalent to (4). {{further|[[equivalence of definitions of alternative ring]]}}
-* The equivalence with definition (4) is [[Artin's theorem on alternative rings]].
+* The equivalence with definition (5) is [[Artin's theorem on alternative rings]].
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Associative ring]] || associativity holds universally || || || {{intermediate notions short|alternative ring|associative ring}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Left-alternative ring]] || || || ||
+|-
+| [[Stronger than::Right-alternative ring]] || || || ||
+|-
+| [[Stronger than::Flexible ring]] || || || ||
+|-
+| [[Stronger than::Power-associative ring]] || || || ||
+|}