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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 said to be right (countably) s-CS if any direct sum of (countably many) copies of the right R-module R is CS. It is shown that for a right countably s-CS ring R the following are equivalent: (i) R is right s-CS, (ii) R has ACC or DCC on projective principal right ideals, (iii) R has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of R, (iv) R is semiperfect. From results of Oshiro [ 12], [13], under these conditions, R is left artinian and QF-3. As a consequence, a ring R is quasi-Frobenius if it is right countably s-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.

A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-3