Rings whose cyclics are essentially embeddable in projective modules

Abstract

A ring R is quasi-Frobenius (QF) if and only if every right R-module is embeddable in a projective module. We call a ring R a right (left) CEPring if each cyclic right (left) R-module is essentially embeddable in a projective module. Examples of right CEP-rings include QF-rings and right uniserial rings. Indeed R is a QF-ring if and only if R is both a right and left CEP-ring [S]. A right CEP-ring which is QF-3 is shown to be QF (Theorem 3.3). Semiperfect CEP-rings and rings, each of whose homomorphic images is a right CEP-ring, are characterized in Theorems 5.2 and 6.2, respectively. The last section deals with split extensions of right uniserial rings as examples of CEP-rings.