RINGS OF OPERATORS ON MODULES OVER COMMUTATIVE RINGS AND THEIR RIGHT IDEALS

Suppose that there is an inclusion of k-algebras R subset of or equal to E subset of or equal to End(k)M with R commutative and E non-commut ative. We introduce and impose conditions under which the finitely gen erated essential right ideals of E may be classified in terms of k-sub modules of M. This yields a classification of the domains Morita equiv alent to E when E is a Noetherian domain. For example, a special case of our results is: THEOREM. Let R be a commutative Noetherian k-algebr a which is domain. Let E be a simple Ore extension of R of the form R[ x,x(-1);sigma] or R[x;delta] (in the latter case we must also assume R superset of Q). Then, for a certain sublattice of the lattice of k-su bmodules of R: (a) Every non-zero right ideal of E is isomorphic to on e of the form E(R, V) = {theta epsilon E: theta(R) subset of or equal to V}, for some V epsilon L. (b) Every domain Morita equivalent to E i s isomorphic to E(V) = {theta epsilon E x Free R: theta(V) subset of o r equal to V}, for some V epsilon L. Conversely, if R is Dedekind, the n E(V) is Morita equivalent to E, for V epsilon L. (C) 1996 Academic P ress, Inc.