\(C\)-injectivity and \(C\)-projectivity. (Q932907)

The author, in this paper, introduces the notions of c-injectivity and c-projectivity for modules and studies their properties. The author studies c-injectivity in relation to Noetherian rings, principal ideal rings and quasi-Frobenius rings. The author gives partial answer to the Matlis problem that ``is every direct summand of a completely decomposable module completely decomposable?''. This he proves under the additional hypothesis that the ring \(A\) is left nonsingular such that every direct sum of injective hulls of cyclic singular modules is injective and every singular left \(A\)-module is c-injective. The author proves that every cyclic c-injective (resp. c-projective) is injective (resp. projective). The proof is standard and simple and hence should have been avoided. The author proves that over a von Neumann regular ring every c-projective module is projective (in this proof, the author for the result that ``over a left nonsingular ring, an essential extension of a singular module is singular'' has given his earlier paper as reference but \textit{K. R. Goodearl}'s book ``Nonsingular rings and modules'' (1976; Zbl 0336.16001) should have been proper reference as the latter is older than the former) and then gets a characterization of von Neumann regular, hereditary rings in terms of c-projectivity. In Section 4, the author generalizes a result of Levy. Finally, the author gets characterizations of semisimple Artinian rings and self-injective regular rings in terms of c-projectivity.