A decomposition theorem for \(\wp^*\)-semisimple rings.
An associative ring \(R\) with identity is defined to be right (left) \(\wp^*\)-semisimple if every right (left) \(R\)-module is the direct sum of a projective module and a quasi-continuous module. As is pointed out in the paper, right \(\wp^*\)-semisimple rings are right Artinian, but need not be left \(\wp^*\)-semisimple. In fact, right \(\wp^*\)-semisimple rings are precisely those rings \(R\) over which every right \(R\)-module is the direct sum of a projective module and a quasi-injective module. The present paper discusses rings which are both right and left \(\wp^*\)-semisimple. The main result of the paper is a structure theorem of a right and left \(\wp^*\)-semisimple ring \(R\), representing \(R\) as a ring-theoretic direct sum of five rings satisfying some specific conditions. As a consequence, \(R\) is \(\wp^*\)-semisimple if and only if there is a ring-direct decomposition \(R=D\oplus E\), where every right (and left) \(D\)-module is the direct sum of a projective module and a semisimple module, and every right (and left) \(E\)-module is the direct sum of a projective module and an injective module.
scientific article; zbMATH DE number 2032647
scientific article