Associated Lie ring of an associative ring

Definition

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 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