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 $G$.