Rings of operators on modules over commutative rings and their right ideals (Q1125888)

\textit{S. P. Smith} and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 56, No. 2, 229-259 (1988; Zbl 0672.14017)] gave examples of domains Morita equivalent to \({\mathcal D}(R)\), the ring of differential operators on the coordinate ring \(R\) of a smooth affine curve. They proved that \({\mathcal D}(R)\overset {M}\sim{\mathcal D}(S)\), for \(S\) a subalgebra of \(R\) with integral closure \(R\) and such that \(\text{Spec }R\to\text{Spec }S\) is injective. The authors [J. Algebra 167, No. 1, 116-141 (1994; Zbl 0824.16022)] classified the right ideals of \({\mathcal D}(R)\) up to isomorphism and the domains Morita equivalent to \({\mathcal D}(R)\). In order to generalise their earlier results the authors now consider a domain \(R\) which is a commutative Noetherian \(k\)-algebra and a simple Ore extension \(E\) of the form \(R[x,x^{-1};\sigma]\) or \(R[x;\delta]\) (in the latter case they assume \(R\supseteq\mathbb{Q}\)). Let \({\mathcal I}(E_E)\) denote the lattice of right ideals of \(E\) which contain a non-zero ideal of \(R\) and \({\mathcal I}(R_k)\) the lattice of \(k\)-submodules of \(R\) which contain some non-zero ideal of \(R\). Given \(V\in{\mathcal I}(R_k)\) let \(E(R,V)=\{\theta\in E:\theta*R\subseteq V\}\) and given \(D\in{\mathcal I}(E_E)\) let \(D*R=\{\sum_id_i*r_i:d_i\in D\), \(r_i\in R\}\), where \(*\) denotes evaluation. The non-zero elements of \(R\) form an Ore set in \(E\) and so one can form the ring \(E\otimes\text{Frac }R\) which acts naturally on \(\text{Frac }R\), and for any \(V\in{\mathcal I}(R_k)\), \(E(V)\) denotes the subring \(\{\theta\in E\otimes\text{Frac }R:\theta*V\subseteq V\}\). The authors prove that the mapping \(-*R:{\mathcal I}(E_E)\to{\mathcal I}(R_k)\) is injective and that \(D\in E(R,D*R)\) for all \(D\in{\mathcal I}(E_E)\). In consequence, for each non-zero right ideal \(D\) of \(E\) there exists \(V\in{\mathcal I}(E_E)*R\) such that \(D\cong E(R,V)\). Moreover, if \(S\) is a domain Morita equivalent to \(E\) then \(S\) is isomorphic to \(E(V)\) for some \(V\in {\mathcal I}(E_E)*R\) and the converse holds if \(R\) is Dedekind.