Δ-locally nilpotent algebras, their ideal structure and simplicity criteria

The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings D(A) of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of D(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[x1,...,xn;δ1,...,δn], certain generalized Weyl algebras, and others. In [8], simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60’th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of [8]. Examples are considered.