Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all n. If U is such and algebra which contains a finitely generated commutative subalgebra A, then we show that any A-torsion theory defined by the coheight of prime ideals is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in Spec A have the same coheight. Hence,thecoheight of the associated prime ideals of A is an invariant of a given simple U-module. This implies a stratification of the category of $U$-modules controlled by the coheight of associated prime ideals of A. Our approach can be viewed as a generalization of the classical paper by R.Block, it allows in particular to study representations of gl(n) beyond the classical category of weight or generalized weight modules.