Rings of Operators on Modules over Commutative Rings and Their Right Ideals Original Research Article

Suppose that there is an inclusion ofk-algebrasRsubset of or equal toEsubset of or equal toEndkMwithRcommutative 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: Timage.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 Rsuperset of3).Then,for a certain sublatticeimageof 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 Vset membership, variantimage. (b)Every domain Morita equivalent to E is isomorphic to[formula]for some Vset membership, variantimage.Conversely,if R is Dedekind,then E(V)is Morita equivalent to E,for Vset membership, variantimage.