On maximal submodules of a finite direct sum of hollow modules. III (Q2266104)

In order to get a generalization of right artinian serial rings the author studied right artinian rings R, with \(J^ 2=0\) \((J=Jacobson\) radical of R), satisfying condition (I): every submodule in any finite direct sum of hollow modules is also a direct sum of hollow modules. [cf. Part I, II; ibid. 21, 649-670, 671-677 (1984; Zbl 0543.16007 and Zbl 0543.16008)]. In this paper, new conditions are introduced: (II'): every hollow module is quasi-projective, (II''): R is an algebra of finite dimension over an algebraically closed field. - It is shown that if R satisfies (I) and either (II') or (II'') then \(| eJ/eJ^ 2| \leq 2\) for each primitive idempotent e. It is conjectured that (II): \(| eJ/eJ^ 2| \leq 2\) for each primitive idempotent e holds without assuming \(J^ 2=0\). The author characterizes rings satisfying (I) and (II) with the additional restriction that \(J^ 3=0\).