Ideal of a Lie ring

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 $$y \in A$$.

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 $$G$$.

Stronger properties

 * Weaker than::Derivation-invariant Lie subring

Weaker properties

 * Stronger than::Lie subring whose sum with any subring is a subring
 * Stronger than::Sub-ideal of a Lie ring