Fully invariant ideal of a Lie ring

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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

Suppose L is a Lie ring and S is a subring of L. We say that S is a fully invariant ideal of L if S is an ideal of L and S is a fully invariant Lie subring of L, i.e., it is invariant under all the Lie ring endomorphisms of L.

Facts

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant derivation-invariant Lie subring fully invariant and derivation-invariant via derivation-invariant implies ideal |FULL LIST, MORE INFO
fully invariant subgroup of additive group of a Lie ring (via fully invariant derivation-invariant Lie subring) Fully invariant derivation-invariant Lie subring|FULL LIST, MORE INFO
verbal ideal of a Lie ring |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant Lie subring fully invariant, but just a subring and not necessarily an ideal |FULL LIST, MORE INFO
characteristic ideal of a Lie ring ideal that is invariant under all automorphisms |FULL LIST, MORE INFO
characteristic Lie subring subring that is invariant under all automorphisms Characteristic ideal of a Lie ring|FULL LIST, MORE INFO