# Derived subring of a Lie ring

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## Definition

Let be a Lie ring. The **derived subring** or **commutator subring** of , denoted , or is defined in the following ways:

- It is the additive subgroup generated by all elements of the form , where
- It is the Lie subring generated by all elements of the form , where
- It is the Lie ideal generated by all elements of the form , where

In situations where the Lie ring is an algebra over some field or ring, the derived subring is also a subalgebra and an ideal over that field or ring. In those cases, it may be termed the **derived subalgebra** or **commutator subalgebra**.