Derivation-invariance is transitive: Difference between revisions

Latest revision as of 22:16, 27 June 2012

This article gives the statement, and possibly proof, of a Lie subring property (i.e., derivation-invariant Lie subring) satisfying a Lie subring metaproperty (i.e., transitive Lie subring property)
View all Lie subring metaproperty satisfactions | View all Lie subring metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for Lie subring properties
Get more facts about derivation-invariant Lie subring |Get facts that use property satisfaction of derivation-invariant Lie subring | Get facts that use property satisfaction of derivation-invariant Lie subring|Get more facts about transitive Lie subring property

ANALOGY: This is an analogue in Lie rings of a fact encountered in group. The old fact is: characteristicity is transitive.
Another analogue to the same fact, in the same new context, is: characteristicity is transitive for Lie rings
View other analogues of characteristicity is transitive|View other analogues from group to Lie ring (OR, View as a tabulated list)

Statement

A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring. In symbols, if $L$ is a Lie ring with subrings $A, B$ such that $B$ is a derivation-invariant Lie subring of $L$ and $A$ is a derivation-invariant Lie subring of $B$ , then $A$ is a derivation-invariant Lie subring of $L$ .

Related facts

Proof

Given: A Lie ring $L$ with Lie subrings $A \leq B \leq L$ . $B$ is a derivation-invariant Lie subring of $L$ and $A$ is a derivation-invariant subring of $B$ .

To prove: $A$ is a derivation-invariant subring of $L$ .

Proof: Suppose $d$ is a derivation of $L$ .

Since $B$ is a derivation-invariant subring of $L$ , $d$ restricts to a map from $B$ to itself. Let $d^{'}$ be the restriction of $d$ to $B$ . Clearly, $d^{'}$ is a derivation of $B$ .

Since $A$ is derivation-invariant in $B$ , $d^{'}$ restricts to a map from $A$ to itself. Thus, $d$ restricts to a map from $A$ to $A$ .

@@ Line 6: / Line 6: @@
 new generic context = Lie ring|
 old fact = characteristicity is transitive|
-alternative analogue = characteristicity is transitive}}
+alternative analogue = characteristicity is transitive for Lie rings}}
 ==Statement==
-A [[derivation-invariant Lie subring]] of a derivation-invariant Lie subring is a derivation-invariant Lie subring.
+A [[derivation-invariant Lie subring]] of a derivation-invariant Lie subring is a derivation-invariant Lie subring. In symbols, if <math>L</math> is a Lie ring with subrings <math>A,B</math> such that <math>B</math>is a derivation-invariant Lie subring of <math>L</math> and <math>A</math> is a derivation-invariant Lie subring of <math>B</math>, then <math>A</math> is a derivation-invariant Lie subring of <math>L</math>.
 ==Related facts==
 * [[Derivation-invariant subring of ideal implies ideal]]
+* [[Invariance under any derivation with partial divided Leibniz condition powers is transitive]]
 ==Proof==
-'''Given''': A Lie ring <math>L</math> with Lie subrings <math>A \le B \le L</math>. <math>B</math> is a derivation-invariant Lie subring of <math>L</math>.
+'''Given''': A Lie ring <math>L</math> with Lie subrings <math>A \le B \le L</math>. <math>B</math> is a derivation-invariant Lie subring of <math>L</math> and <math>A</math> is a derivation-invariant subring of <math>B</matH>.
 '''To prove''': <math>A</math> is a derivation-invariant subring of <math>L</math>.