Abstract

A module M is said to satisfy the condition if M is a direct sum of a projective module and a quasi-continuous module. By Huynh and Rizvi (J. Algebra 223 (2000) 133; Characterizing rings by a direct decomposition property of their modules, preprint 2002) rings over which every countably generated right module satisfies are exactly those rings over which every right module is a direct sum of a projective module and a quasi-injective module. These rings are called right -semisimple rings. Right -semisimple rings are right artinian. However, in general, a right -semisimple rings need not be left -semisimple. In this note, we will prove a ring-direct decomposition theorem for right and left -semisimple rings. Moreover, we will describe the structure of each direct summand in the obtained decomposition of these rings.