Small quasi-projective modules

We study small quasi-projective modules and closely related classes of modules. The concept of a small quasi-projective module is dual to the concept of an essentially quasi-injective module, which has recently been studied in several works. It is shown that over right perfect rings, the class of small quasi-projective right modules coincides with a number of classes of right modules close to projective modules, which are studied in the article. As a consequence of the obtained results, the well-known A.A. Tuganbaev’s theorem on the coincidence of the classes of quasi-projective right modules and endomorphism-lifting right modules over right perfect rings is presented. Also, characterizations are obtained for modules M , for which in the category σ [ M ] every (finitely generated, cyclic, semisimple, simple) module is small projective in σ [ M ] .