A decomposition theorem for *-semisimple rings

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.