Difference between revisions of "Derivation-invariance is transitive"

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old generic context = group|
 
old generic context = group|
 
new generic context = Lie ring|
 
new generic context = Lie ring|
old fact = characteristicity is transitive}}
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old fact = characteristicity is transitive|
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alternative analogue = characteristicity is transitive for Lie rings}}
 
==Statement==
 
==Statement==
  
A [[derivation-invariant Lie subring]] of a derivation-invariant Lie subring is a derivation-invariant Lie subring.
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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==
 
==Related facts==
  
 
* [[Derivation-invariant subring of ideal implies ideal]]
 
* [[Derivation-invariant subring of ideal implies ideal]]
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* [[Invariance under any derivation with partial divided Leibniz condition powers is transitive]]
  
 
==Proof==
 
==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>.
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'''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>.
 
'''To prove''': <math>A</math> is a derivation-invariant subring of <math>L</math>.

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 Bis 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 \le B \le 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.