Lie subring whose sum with any subring is a subring

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: permutable subgroup
View other analogues of permutable 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 Lie subring whose sum with any subring is a subring if, for any Lie subring $A$ of $L$, the subgroup $S+A$ is also a Lie subring of $L$.

Relation with other properties

Stronger properties

• Ideal of a Lie ring: For proof of the implication, refer Sum of ideal and subring is subring and for proof of its strictness (i.e. the reverse implication being false) refer Lie subring whose sum with any subring is a subring not implies ideal.
• Subring of a Lie ring that is maximal as a subgroup

Metaproperties

A join of Lie subrings with this property also has this property.