Simplicity criteria for rings of differential operators.

Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal}, $\CD (\CA )$ be the algebra of differential operators on the algebra $\CA$. The aim of the paper is to give a simplicity criterion for the algebra $\CD (\CA )$: {\em The algebra $\CD (\CA )$ is simple iff $\CD (\CA ) \ga_r^i\CD (\CA )= \CD (\CA )$ for all $i\geq 1$ provided the field $K$ is a perfect field.} Furthermore, a simplicity criterion is given for the algebra $\CD (R)$ of differential operators on an arbitrary commutative algebra $R$ over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.