  The class of $\D$-locally nilpotent algebras (introduced in the paper) is a
wide generalization of the algebras of differential operators on commutative
algebras. Examples includes all the rings $\CD (A)$ of differential operators
on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD
(A)$ that contain the algebra $A$, the universal enveloping algebras of
nilpotent, solvable and semi-simple Lie algebras, the Poisson universal
enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions
$A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl
algebras, and others.
  In \cite{SimCrit-difop}, simplicity criteria are given for the algebras
differential operators on commutative algebras (it was a long standing
problem). The aim of the paper is to describe the ideal structure of
$\D$-locally nilpotent algebras and as a corollary to give simplicity criteria
for them (it is a generalization of the results of \cite{SimCrit-difop}).
Examples are considered.
