A characterization of co-Harada ring. (Q2853416)

A submodule \(N\) of a module \(M\) is called small (or superfluous) if for any proper submodule \(P\) of \(M\), \(N+P\) is a proper submodule of \(M\). This concept is the dual of the concept ``essential submodule'' (note, a nonzero submodule \(N\) of \(M\) is essential if it intersects every nonzero submodule of \(M\) nontrivially). A module \(M\) is called small if it is a small submodule of its injective hull. It's easy to see that if \(M\) is a non-small module, it is a non-small submodule in any extension module of \(M\). Dually, \(M\) is called a non-cosmall module if \(M\) is a homomorphic image of a projective module \(P\) whose kernel is not essential in \(P\). A ring \(R\) is called left Harada if it is left Artinian and every non-small module contains a nonzero injective module. Dually, \(R\) is called a co-Harada ring if it has the ascending chain condition (acc, for brevity) on right annihilators and every non-cosmall module contains a nonzero projective direct summand. In [\textit{K. Oshiro}, Hokkaido Math. J. 13, 310-338 (1984; Zbl 0559.16013)], it is shown that left Harada rings and right co-Harada rings coincide (left Harada rings and right Harada rings may not coincide, in general).NEWLINENEWLINE The main result (Theorem 7) of the article under review shows that if \(R\) is a left perfect ring with the acc on right annihilators such that every essential extension module of \(R_R\) is projective, then \(R\) is a left Harada ring. The converse of this fact is well-known, [see loc. cit., Theorem 3.18]. The reviewer observes trivially that one may trade off the property of the left ``perfectness'' in the statement of Lemma 3, with a weaker property, namely, the left ``semi-Artinianness'' and the proof remains intact.