A simplified proof of Oshiro's theorem for co-\(H\) rings (Q1293408)

A ring \(R\) is called a right co-H-ring if every projective right \(R\)-module has the property that its submodules are essential in direct summands. Right co-H-rings present a generalization of both QF-rings and Nakayama rings. \textit{K. Oshiro} [Hokkaido Math. J. 13, 310-338 (1984; Zbl 0559.16013)] has shown that the following conditions are equivalent for a ring \(R\): 1) \(R\) is a right co-H-ring; 2) Every right \(R\)-module is a direct sum of a projective module and a singular module; 3) The class of projective right \(R\)-modules is closed under essential extensions; 4) \(R\) has ACC on right annihilators and every right \(R\)-module which is not singular contains a nonzero projective direct summand (\(R\) is in fact a semiprimary QF-3-ring). In the paper under review, the author provides a simplified proof of the above result.