  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.
