Abelian Lie ring

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.

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 $$G$$.

Stronger properties

 * Weaker than::Cyclic Lie ring

Weaker properties

 * Stronger than::Nilpotent Lie ring
 * Stronger than::Solvable Lie ring