On the torsion theories of Morita equivalent rings (Q1925022)

Let \(R\) and \(S\) be Morita equivalent rings, and let \(W\) be a bimodule that gives the equivalence of categories: \(\alpha=W\otimes_R-:R\text{-mod}\to S\text{-mod}\). Let \(0\to R\to E_0\to E_1\to\cdots\) be a minimal injective resolution of \(R\), and let \(\sigma_n\) be the torsion theory cogenerated by \(E_0\oplus E_1\oplus\cdots\oplus E_n\). Let \(\tau_n\) be a torsion theory defined in a similar way by a minimal injective resolution of \(S\). Then \(\alpha(\sigma_n(R))=\tau_n(S)\), and hence the localizations \(R_{\sigma_n}\) and \(S_{\tau_n}\) are Morita equivalent. This paper also examines the effect of \(\alpha\) on properties of some other torsion theories, especially the Goldie torsion theory.