Quasi-projective covers and direct sums

Abstract:

In this paper $R$ denotes an associative ring with an identity, and all modules are unital left $R$-modules. It is shown that the existence of a quasi-projective cover for each module implies that each module has a projective cover. By a similar technique the following statements are shown to be equivalent: 1. $R$ is semisimple and Artinian; 2. Every finitely generated module is quasi-projective; and 3. The direct sum of every pair of quasi-projective modules is quasi-projective. Direct sums of quasi-injective modules are also investigated.