Rings whose cyclics are essentially embeddable in projective modules (Q912190)

The authors call a ring \(R\) a right (left) CEP-ring if each cyclic right (left) \(R\)-module is essentially embeddable in a projective module. Quasi-Frobenius (QF) rings and right uniserial rings are examples of right CEP-rings and in fact a theorem of \textit{C. Faith} and \textit{E. A. Walker} [J. Algebra 5, 203-221 (1967; Zbl 0173.03203)] shows that \(R\) is QF if and only if \(R\) is both a right and a left CEP-ring. Here it is shown that if \(R\) is both a QF-3 ring and a right CEP-ring then \(R\) is QF. (Recall that \(R\) is said to be QF-3 if its right injective hull is projective.) Moreover it is shown that a semiperfect ring \(R\) is right CEP if and only if (i) \(R\) is right Artinian and (ii) every indecomposable projective module is both uniform and weakly \(R\)-injective. (Here a module \(P\) is called weakly \(R\)-injective if, given any \(f\in\Hom(R,E(P))\), where \(E(P)\) is the injective hull of \(P\), then \(f(1)\) belongs to some submodule \(X\) of \(E(P)\) isomorphic to \(P\). Also, as a ``Note added in proof'', the authors say that the uniform assumption may be omitted in this characterization.) Another main result characterizes those rings \(R\) of which every homomorphic image is a right CEP-ring as being a direct sum of right uniserial rings or matrix rings over right self-injective right uniserial rings. The paper ends with illustrative examples obtained using the trivial (split) extension construction.