# Difference between revisions of "Derivation-invariance is transitive"

(→Related facts) |
|||

(4 intermediate revisions by the same user not shown) | |||

Line 5: | Line 5: | ||

old generic context = group| | old generic context = group| | ||

new generic context = Lie ring| | new generic context = Lie ring| | ||

− | old fact = characteristicity is transitive}} | + | old fact = characteristicity is transitive| |

+ | 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. | + | 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]] | ||

+ | * [[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>. | + | '''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 is a Lie ring with subrings such that is a derivation-invariant Lie subring of and is a derivation-invariant Lie subring of , then is a derivation-invariant Lie subring of .

## 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 with Lie subrings . is a derivation-invariant Lie subring of and is a derivation-invariant subring of .

**To prove**: is a derivation-invariant subring of .

**Proof**: Suppose is a derivation of .

Since is a derivation-invariant subring of , restricts to a map from to itself. Let be the restriction of to . Clearly, is a *derivation* of .

Since is derivation-invariant in , restricts to a map from to itself. Thus, restricts to a map from to .