Right perfect rings with the extending property on finitely generated free modules (Q913920)

Let R be a ring with identity. If R satisfies the condition: \((^*)^*\) Every non-cosmall right R-module contains a nonzero projective direct summand [cf. \textit{M. Harada}, Proc. 1978 Antwerp Conf., Lect. Notes Pure Appl. Math. 51, 669-690 (1979; Zbl 0449.16018)] and ACC on right annihilator ideals, then it is called a right co-H-ring by \textit{K. Oshiro} [Hokkaido Math. J. 13, 310-338 (1984; Zbl 0559.16013)]. A right R-module M is said to be an extending module if for any submodule A of M there is a direct summand \(A^*\) of M containing A such that A is essential in \(A^*\). Using this concept, the author first proves the following theorem. Theorem I. A right perfect ring R satisfies \((^*)^*\) iff \(R^ k_ R\) is an extending module for each \(k>0\) iff \(R^ 2_ R\) is an extending module. He then obtains the following characterization of right co-H-rings. Theorem II. A ring R is a right co- H-ring iff (1) R is right perfect, (2) R satisfies ACC on right annihilator ideals and (3) \(R^ 2_ R\) is an extending module. He also considers the case where R is right non-singular.