Characteristic ideal of a Lie ring: Difference between revisions

Latest revision as of 05:18, 27 July 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
View other analogues of characteristic subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

A subset of a Lie ring is termed a characteristic ideal if it satisfies both the following conditions:

It is a characteristic Lie subring, i.e., it is invariant under all the automorphisms of the whole Lie ring.
It is an ideal of the Lie ring, i.e., it is invariant under all the inner derivations of the whole Lie ring.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant ideal of a Lie ring	ideal that is invariant under all Lie ring endomorphisms			\|FULL LIST, MORE INFO
verbal ideal of a Lie ring	ideal that is generated by a set of words.			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic Lie subring	invariant under all Lie ring automorphisms	characteristic subring not implies ideal		\|FULL LIST, MORE INFO
ideal of a Lie ring	invariant under all inner derivations			\|FULL LIST, MORE INFO

@@ Line 1: / Line 1: @@
-{{Lie subalgebra property}}
+{{Lie subring property}}
-{{Lie algebra analogue of subgroup property|[[characteristic subgroup]]}}
+{{analogue of property|
+new generic context = Lie ring|
+new specific context = Lie subring|
+old generic context = group|
+old specific context = subgroup|
+old property = characteristic subgroup}}
 ==Definition==
-A subset of a Lie algebra is termed a '''charcateristic ideal''' if it satisfies the following equivalent conditions:
+A subset of a [[Lie ring]] is termed a '''characteristic ideal''' if it satisfies '''both''' the following conditions:
-* It is a [[subalgebra of a Lie algebra|subalgebra]] and it is invariant under all automorphisms of the Lie algebra
+# It is a [[characteristic Lie subring]], i.e., it is invariant under all the automorphisms of the whole Lie ring.
-* It is an [[ideal of a Lie algebra|ideal]] and it is invariant under all automorphisms of the Lie algebra
+# It is an [[ideal of a Lie ring|ideal]] of the Lie ring, i.e., it is invariant under all the [[inner derivation]]s of the whole Lie ring.
+==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::fully invariant ideal of a Lie ring]] || [[ideal of a Lie ring|ideal]] that is invariant under all Lie ring endomorphisms || || || {{intermediate notions short|characteristic ideal of a Lie ring|fully invariant ideal of a Lie ring}}
+|-
+| [[Weaker than::verbal ideal of a Lie ring]] || [[ideal of a Lie ring|ideal]] that is generated by a set of words. || || || {{intermediate notions short|characteristic ideal of a Lie ring|verbal ideal of a Lie ring}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::characteristic Lie subring]] || invariant under all Lie ring automorphisms || [[characteristic subring not implies ideal]] || || {{intermediate notions short|characteristic Lie subring|characteristic ideal of a Lie ring}}
+|-
+| [[Stronger than::ideal of a Lie ring]] || invariant under all [[inner derivation]]s || || || {{intermediate notions short|ideal of a Lie ring|characteristic ideal of a Lie ring}}
+|}