Semisimple rings of quotients in Morita contexts
Let \((R,{_ R U_ S},{_ S V_ R},S)\) be a Morita context and \((\tau,\tau')\) a pair of corresponding Gabriel topologies, containing trace ideals \(I\) and \(J\). Conditions are investigated for the ring of quotients \(\widehat S\) of \(S\) with respect to \(\tau'\) to be semisimple artinian, simple artinian or a division ring. The main results are the following. 1) \(\widehat S\) is semisimple \(\Leftrightarrow \text{Sat}_ \tau(_ RU)\) satisfies a.c.c. and \(\tau\)- closed modules are injective \(\Leftrightarrow\text{Sat}_ \tau(_ RU)\) satisfies a.c.c. and essential submodules of a \(\tau\)-free module \(M\) are \(\tau\)-dense in \(M \Leftrightarrow\) the lattice \(L(_ RU)\) of submodules of \(_ RU\) is \(\tau\)-cofinally finite and \(\tau\)-closed modules are injective. 2) Suppose that \(d(U/T_ \tau(U)) < \infty\) and \(\widehat R\) is semisimple. Then \(\widehat R\) and \(\widehat S\) are Morita equivalent, hence \(\widehat S\) is semisimple. \(\widehat S\) is simple \(\Leftrightarrow \widehat R\) is simple. 3) Suppose that \(\widehat S\) is semisimple. \(\widehat S\) is simple \(\Leftrightarrow T_{\tau_ 1}(U) = T_ \tau(U)\) for any \(\tau_ 1 \geq\tau \Leftrightarrow \widehat{U} = \tau_ 1\widehat{U}\) for any \(\tau_ 1 \geq \tau\). 4) \(\widehat S\) is a division ring \(\Leftrightarrow{_ RU}\) is \(\tau\)-simple (i.e. \(_ RU\) has only two \(\tau\)-closed submodules: \(T_ \tau(M)\) and \(M\)). 5) Suppose the context is nondegenerate and \(\tau'\) is the dense topology on \(S\)-Mod. \(\widehat{S}\) (the maximal left quotient ring of \(S\)) is semisimple \(\Leftrightarrow Z(_ RU) = 0\) and \(d(_ RU) < \infty\).
scientific article; zbMATH DE number 561567
scientific article