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

Definition

A subset $B$ of a Lie ring $A$ is termed an ideal in $A$ if $B$ is additively a subgroup and $[x,y] \in B$ whenever $x \in B$ and $y \in A$.

Relation with properties in related algebraic structures

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 $R$ is an associative ring: an abelian group with a distributive associative multiplication. We can define the Lie ring associated with $R$ as $R$ with the same addition and wit hthe Lie bracket given by the commutator operation $[x,y] = xy - yx$.

Then, the property of being an ideal of the Lie ring is equivalent to the property of being a Lie ideal in $R$. 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 $G$ is a Lazard Lie group and $L$ is its Lazard Lie ring. Under the natural bijection from $L$ to $G$, the ideals of $L$ correspond to the normal subgroups of $G$.