Ring of differential operators on a Lie ring

This article provides a semi-basic definition in the following area: ring theory

Definition

Suppose $L$ is a Lie ring, and $\mathrm{D}\mathrm{e}\mathrm{r}(L)$ is the Lie ring of derivations of $L$. The ring of differential operators of $L$ is the collection of all maps from $L$ to $L$ that can be expressed as sums and differences of the identity map and composites of derivations. This is a unital ring where:

• Addition is pointwise.
• Multiplication is by composition.
• The zero element is the zero map.
• The multiplicative identity is the identity map.

Elements of this ring are termed differential operators on the Lie ring.

Note that the distributivity follows from the fact that all differential operators are endomorphisms of the underlying abelian group structure.