$\D$-locally nilpotent algebras, their ideal structure and simplicity criteria

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.