Maximal submodules and locally perfect rings
A ring is a right Bass ring if all right modules have maximal submodules. Let \(R\) be a ring with center \(C\). The ring \(R\) is locally \(X\) if the localization \(R_M\) has property \(X\) for every maximal ideal \(M\) of \(C\). The main theorem is a long list of equivalent conditions for a locally module-finite ring \(R\) to be a right Bass ring. These include (1) \(R\) is a left Bass ring, (2) \(R\) is locally a right Bass ring, (3) \(R\) is locally a right perfect ring, (4) \(C\) is a locally perfect ring, (5) \(C/J(C)\) is a von Neumann regular ring and \(J(C)\) is left vanishing where \(J(C)\) is the Jacobson radical of \(C\), and (5) \(R\) is a strongly \(\pi\)-regular ring and \(J(R)\) is left vanishing. These results are all correct. However Corollary 4 and Corollary 5 are not correct. In both of these it is asserted that if \(R\) is a locally module-finite right Bass ring which is semiprime, then \(R\) is von Neumann regular. A counterexample can be constructed as follows. Let \(C\) be a commutative von Neumann regular ring with no minimal ideal and \(M\) a maximal ideal. Let \(R\) be the matrix ring of the form \(\left(\begin{smallmatrix} C&C\\ M&C\end{smallmatrix}\right)\). The ring \(R\) is locally module-finite locally perfect and hence is right Bass. Also \(J(R)=0\) but \(R\) is not von Neumann regular.
scientific article; zbMATH DE number 1274222