Simplicity criteria for rings of differential operators

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