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A module M is called a CS-module if every submodule of M is essential in a direct summand of M. A ring R is called CS-semisimple if every right R-module is CS. For a ring R, we show that: Ž .1 R is right artinian with Jacobson radical cube zero if every countably generated right R-module is a direct sum of a projective module and a CS-module. Ž .Ž . 2 The following conditions are equivalent: i Every countably generated right R-module is a direct sum of a projective module and a quasicontinuous Ž . module; and ii every right R-module is a direct sum of a projective module and a quasi-injective module. Ž .We describe the structure of rings in 2 and show that such a ring is not necessarily CS-semisimple. ᮊ 2000 Academic Press R wx 6, 12 for details on CS-modules. Ž Let ဧ be a property of modules over a ring R such as the property of being injective, being CS, or being a direct sum of a projective module 133

Theorems of Osofsky and Kato imply that a right and left self-injective one-sided perfect ring is quasi-Frobenius (= QF). The corresponding question for one-sided self-injective one or two-sided perfect rings remains open, even assuming that the ring is semiprimary. The latter version of the problem is known as Faith's Conjecture (FC). We survey results on QF rings, especially those obtained in connection with FC. We also review various results that provide partial answers to another problem of Faith: Is a right FGF ring necessarily QF? On this topic, we provide a new result, namely that if all factor rings of R are right FGF, then R is QF (Theorem 6.1). In Sec. 7 we review results concerning the question of when a D-ring is QF. Sections 8 and 9 are devoted respectively to IF rings, and to Σ-injective rings and Σ-CS rings.

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