# 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

View a complete list of such propertiesVIEW 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: normal subgroup

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

## Contents

## Definition

A subset of a Lie ring is termed an *ideal* in if is additively a subgroup and whenever and .

### Lie algebra

`Further information: Ideal of a Lie algebra`

A Lie algebra is a Lie ring that is simultaneously (i.e., with the same operations) an algebra over a field. An ideal of a Lie algebra is an ideal of the underlying Lie ring that is also a linear subspace, i.e., it is closed under multiplication by scalars in the field.

### Ring whose commutator operation is the Lie bracket

Suppose is an associative ring: an abelian group with a distributive associative multiplication. We can define the Lie ring associated with as with the same addition and wit hthe Lie bracket given by the commutator operation .

Then, the property of being an ideal of the Lie ring is equivalent to the property of being a Lie ideal in . Being a *Lie ideal* is weaker than being a two-sided ideal. It is incomparable with the property of being a left ideal or being a right ideal. Moreover, a subset that is both a left ideal and a Lie ideal is two-sided ideal. Similarly, a subset that is both a right ideal and a Lie ideal is a two-sided ideal.

### Group via the Lazard correspondence

Suppose is a Lazard Lie group and is its Lazard Lie ring. Under the natural bijection from to , the ideals of correspond to the normal subgroups of .