Rings of Operators on Modules over Commutative Rings and Their Right Ideals

Abstract

Suppose that there is an inclusion ofk-algebrasR⊆E⊆EndkMwithRcommutative andEnon-commutative. We introduce and impose conditions under which the finitely generated essential right ideals ofEmay be classified in terms ofk-submodules ofM. This yields a classification of the domains Morita equivalent toEwhenEis a Noetherian domain. For example, a special case of our results is:THEOREM.Let R be a commutative Noetherian k-algebra which is domain. Let E be a simple Ore extension of R of the form R[x,x−1;σ]or R[x;δ] (in the latter case we must also assume R⊃3).Then,for a certain sublatticeLof the lattice of k-submodules of R:(a)Every non-zero right ideal of E is isomorphic to one of the form[formula]for some V∈L.(b)Every domain Morita equivalent to E is isomorphic to[formula]for some V∈L.Conversely,if R is Dedekind,then E(V)is Morita equivalent to E,for V∈L.