Ring of differential operators

Summary

The ring of differential operators $D(A)$ of a commutative algebra $A$ over a field $k$ is a ring of $k$-linear maps $A \to A$. Its elements behave like linear differential operators of finite order. Grothendieck's definition uses only commutators with multiplication maps, so it makes sense for any commutative algebra, without coordinates. For the polynomial ring $k[x_1, \dots, x_n]$ in characteristic $0$ it is the Weyl algebra $A_n(k)$. Its modules, called $D$-modules, are a tool for studying linear partial differential equations by algebra.