A decomposition theorem for ℘∗-semisimple rings

AbstractA 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