Fully invariant Lie subring

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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: fully invariant subgroup
View other analogues of fully invariant subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

Definition

A subring S of a Lie ring L is termed a fully invariant Lie subring if, for every endomorphism \sigma of L, \sigma(S) \subseteq S.

Relation with properties in related groups

Lazard Lie ring

Suppose G is a Lazard Lie group and L is a Lazard Lie ring. Under the natural bijection between L and G, fully invariant subrings of L correspond to fully invariant subgroups of G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Fully invariant subgroup of additive group of a Lie ring Fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant
Lie subring invariant under any additive endomorphism satisfying a comultiplication condition

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Characteristic subring of a Lie ring

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive Lie subring property Yes full invariance is transitive for Lie rings If A \le B \le L are Lie rings, with A fully invariant in B and B fully invariant in L, then A is fully invariant in L.
Lie bracket-closed Lie subring property Yes full invariance is Lie bracket-closed for Lie rings If A,B are fully invariant Lie subrings of a Lie ring L, so is [A,B].
strongly intersection-closed Lie subring property Yes full invariance is strongly intersection-closed for Lie rings If A_i, i \in I are all fully invariant Lie subrings of a Lie ring L, then so is \bigcap_{i \in I} A_i.