Characterized rings by pseudo - injective modules

Abstract

It is shown that: (1) Let R be a simple right Noetherian ring, then the following conditions are equivalent: (i) R is a right SI ring; (ii) Every cyclic singular right R - module is pseudo - injective. (2) Let R be a right artinian ring such that every finite generated right R - module is a direct sum of a projective module and a pseudo - injective module. Then: (i) R/Soc(RR) is a semisimple artinian ring; (ii) J(R) Soc(RR); (iii) J2(R) = 0. (3) Let R be a ring with condition ( ), then every singular right R - module is isomorphic with a direct sum of pseudo - injective modules.