Characteristic Lie subring: Difference between revisions

Latest revision as of 16:56, 30 June 2013

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
View a complete list of such properties
VIEW RELATED: Lie subring property implications | Lie subring property non-implications | Lie subring metaproperty satisfactions | Lie subring metaproperty dissatisfactions | Lie subring property satisfactions |Lie subring property dissatisfactions

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie subring property analogous to the subgroup property: characteristic subgroup
An alternative analogue of characteristic subgroup in Lie ring is: derivation-invariant Lie subring
View other analogues of characteristic subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

A subring of a Lie ring is termed a characteristic Lie subring or characteristic subring if it is invariant under all automorphisms of the Lie ring.

Relation with properties in related groups

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
fully invariant Lie subring	Lie subring that is invariant under all endomorphisms	\|FULL LIST, MORE INFO
fully invariant subgroup of additive group of a Lie ring	subset of a Lie ring that is invariant under all endomorphisms of the additive group of the Lie ring	\|FULL LIST, MORE INFO
verbal Lie subring	generated by a set of words using the Lie operations	\|FULL LIST, MORE INFO
marginal Lie subring		\|FULL LIST, MORE INFO
strictly characteristic Lie subring		\|FULL LIST, MORE INFO
injective endomorphism-invariant Lie subring		\|FULL LIST, MORE INFO}
characteristic ideal of a Lie ring	characteristic and also an ideal	\|FULL LIST, MORE INFO
characteristic derivation-invariant Lie subring	characteristic and also a derivation-invariant Lie subring	\|FULL LIST, MORE INFO

Incomparable properties

Property	Meaning	Proof of forward non-implication	Proof of reverse non-implication	Conjunction
ideal of a Lie ring	invariant under all inner derivations	characteristic not implies ideal	ideal not implies characteristic	characteristic ideal of a Lie ring
derivation-invariant Lie subring	invariant under all derivations	characteristic not implies derivation-invariant	derivation-invariant not implies characteristic	characteristic derivation-invariant Lie subring

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive Lie subring property	Yes	characteristicity is transitive for Lie rings	Suppose $A\leq B\leq L$ are Lie rings such that $A$ is a characteristic subring of $B$ and $B$ is a characteristic subring of $L$ . Then, $A$ is a characteristic subring of $L$ .
Lie bracket-closed Lie subring property	Yes	characteristicity is Lie bracket-closed for Lie rings	Suppose $A,B\leq L$ are characteristic subrings. Then, the Lie bracket $[A,B]$ is also a characteristic subring of $L$ .

@@ Line 10: / Line 10: @@
 ==Definition==
-A [[subring of a Lie ring|subring]] of a [[Lie ring]] is termed a '''characteristic subring''' if it is invariant under all automorphisms of the Lie ring.
+A [[subring of a Lie ring|subring]] of a [[Lie ring]] is termed a '''characteristic Lie subring''' or '''characteristic subring''' if it is invariant under all automorphisms of the Lie ring.
 ==Relation with properties in related groups==
@@ Line 17: / Line 17: @@
 * [[Lazard correspondence establishes a correspondence between powering-invariant characteristic subgroups and powering-invariant characteristic subrings]]
-==Metaproperties==
+==Relation with other properties==
-{{transitive Lie subring property}}
+===Stronger properties===
-A characteristic Lie subring of a characteristic Lie subring is characteristic in the whole Lie ring. {{proofat|[[Characteristicity is transitive for Lie rings]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::fully invariant Lie subring]] || Lie subring that is invariant under all endomorphisms || || || {{intermediate notions short|characteristic Lie subring|fully invariant Lie subring}}
+|-
+| [[Weaker than::fully invariant subgroup of additive group of a Lie ring]] || subset of a Lie ring that is invariant under all endomorphisms of the additive group of the Lie ring || || || {{intermediate notions short|characteristic Lie subring|fully invariant subgroup of additive group of a Lie ring}}
+|-
+| [[Weaker than::verbal Lie subring]] || generated by a set of words using the Lie operations || || || {{intermediate notions short|characteristic Lie subring|verbal Lie subring}}
+|-
+| [[Weaker than::marginal Lie subring]] || || || || {{intermediate notions short|characteristic Lie subring|marginal Lie subring}}
+|-
+| [[Weaker than::strictly characteristic Lie subring]] || || || || {{intermediate notions short|characteristic Lie subring|strictly characteristic Lie subring}}
+|-
+| [[Weaker than::injective endomorphism-invariant Lie subring]] || || || || {{intermediate notions short|characteristic Lie subring|injective endomorphism-invariant Lie subring}}}
+|-
+| [[Weaker than::characteristic ideal of a Lie ring]] ||characteristic and also an [[ideal of a Lie ring|ideal]] || || || {{intermediate notions short|characteristic Lie subring|characteristic ideal of a Lie ring}}
+|-
+| [[Weaker than::characteristic derivation-invariant Lie subring]] || characteristic and also a [[derivation-invariant Lie subring]] || || || {{intermediate notions short|characteristic Lie subring|characteristic derivation-invariant Lie subring}}
+|}
-{{Lie bracket-closed Lie subring property}}
+===Incomparable properties===
-The Lie bracket of two characteristic subrings of a Lie ring is again a characteristic subring. {{proofat|[[Characteristicity is Lie bracket-closed for Lie rings]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of forward non-implication !! Proof of reverse non-implication !! Conjunction
+|-
+| [[ideal of a Lie ring]] || invariant under all [[inner derivation of a Lie ring|inner derivation]]s || [[characteristic not implies ideal]] || [[ideal not implies characteristic]] || [[characteristic ideal of a Lie ring]]
+|-
+| [[derivation-invariant Lie subring]] || invariant under all [[derivation of a Lie ring|derivation]]s || [[characteristic not implies derivation-invariant]] || [[derivation-invariant not implies characteristic]] || [[characteristic derivation-invariant Lie subring]]
+|}
-{{intersection-closed Lie subring property}}
+==Metaproperties==
-An arbitrary intersection of characteristic Lie subrings is a characteristic Lie subring.
-{{join-closed Lie subring property}}
-An arbitrary join of characteristic Lie subrings is a characteristic Lie subring.
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[transitive Lie subring property]] || Yes || [[characteristicity is transitive for Lie rings]] || Suppose <math>A \le B \le L</math> are [[Lie ring]]s such that <math>A</math> is a characteristic subring of <math>B</math> and <math>B</math> is a characteristic subring of <math>L</math>. Then, <math>A</math> is a characteristic subring of <math>L</math>.
+|-
+| [[Lie bracket-closed Lie subring property]] || Yes || [[characteristicity is Lie bracket-closed for Lie rings]] || Suppose <math>A,B \le L</math> are characteristic subrings. Then, the Lie bracket <math>[A,B]</math> is also a characteristic subring of <math>L</math>.
+|}