# Abelian Lie ring

From Groupprops

This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.

View a complete list of properties of Lie ringsVIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions

ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: abelian group

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

## Contents

## Definition

An **abelian Lie ring** is a Lie ring satisfying the following equivalent conditions:

- The Lie bracket of any two elements is zero.
- Every Lie subring of the Lie ring is an ideal in the Lie ring.

### Equivalence of definitions

`Further information: Lie ring is abelian iff every subring is an ideal`

### Lie algebra

An abelian Lie algebra is an abelian Lie ring that is also a Lie algebra.

### Ring whose commutator operation is the Lie bracket

Suppose is an associative ring. can be viewed as a Lie ring with the Lie bracket as . The Lie ring is an abelian Lie ring if and only if is a commutative ring.

### Group via the Lazard correspondence

Suppose is a Lazard Lie group and is its Lazard Lie ring. is an abelian Lie ring if and only if is an abelian group.

Moreover, under the natural bijection from to , abelian subrings of correspond to abelian subgroups of .