# 2-sub-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: 2-subnormal subgroup
View other analogues of 2-subnormal subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)

## Definition

A subring of a Lie ring is termed a 2-sub-ideal if it is an ideal of an ideal of the Lie ring.

## Relation with properties in related algebraic structures

### Lie algebra

Further information: 2-sub-ideal of a Lie algebra

A 2-sub-ideal of a Lie algebra is a 2-sub-ideal of a Lie ring that is also a linear subspace.

Note that there is a somewhat subtle but important point here. For the definition of 2-sub-ideal, we need it to be an ideal of an ideal of the whole ring. It turns out that, if the bottom subset is a linear subspace of the Lie algebra, we can choose the intermediate ideal to also be a linear subspace, and hence to be an ideal in the Lie algebra sense.

### Group via the Lazard correspondence

Suppose $G$ is a Lazard Lie group and $L$ is its Lazard Lie ring. Under the natural bijection between $L$ and $G$, the 2-sub-ideals of $L$ correspond to the 2-subnormal subgroups of $G$.