# Ring of differential operators on a Lie ring

## Definition

Suppose is a Lie ring, and is the Lie ring of derivations of . The **ring of differential operators** of is the collection of all maps from to 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.