# Abelian Lie ring

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 rings
VIEW 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)

## Definition

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

1. The Lie bracket of any two elements is zero.
2. 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

## Relation with properties in related algebraic structures

### 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 $R$ is an associative ring. $R$ can be viewed as a Lie ring with the Lie bracket as $[x,y] = xy - yx$. The Lie ring $R$ is an abelian Lie ring if and only if $R$ is a commutative ring.

### Group via the Lazard correspondence

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

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