Associated Lie ring of an associative ring

For rings
Suppose $$L$$ is an associative ring (note that $$L$$ does not need to be unital or commutative). The associated Lie ring of $$L$$ is a Lie ring defined as follows:


 * The underlying set and additive group structure are the same as those of $$L$$.
 * The Lie bracket operation is defined as the defining ingredient::additive commutator for $$L$$, i.e., $$[x,y] := xy - yx$$.

For algebras
The above definition can be adapted to the case that $$L$$ is an associative algebra over a commutative unital ring $$R$$. In this case, we construct the Lie algebra of $$L$$ using the same recipe as above, but we now additionally have a $$R$$-module structure.

Note that the underlying Lie ring structure remains the same regardless of what commutative unital ring we consider $$L$$ as an algebra over.

Facts in the definition

 * Associated Lie algebra of an associative algebra is a Lie algebra

Related notions

 * Associated Malcev ring of an alternative ring