Fully invariant ideal of a Lie ring

This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
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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

Fully invariant implies ideal for class two Lie ring

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	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)	\|FULL LIST, MORE INFO
verbal ideal of a Lie ring			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	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	\|FULL LIST, MORE INFO