  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.
