On maximal submodules of a finite direct sum of hollow modules. IV (Q1063666)

[Part III, cf. the first author, ibid. 81-98 (1985; Zbl 0559.16010).] R is a right artinian ring with 1 and every R-module M is unitary with finite composition length \(| M|\). If M has a unique maximal submodule J(M), M is called hollow. The authors study conditions under which every maximal submodule of a finite direct sum D of certain hollow modules over R contains a non-zero direct summand of D. For any hollow R- module M we have \(M\cong eR/A\) for a primitive idempotent e and a right ideal A in R. In addition, if \(N=eR/A\), then \(\Delta =End_ R(N/J(N))\) is a division ring. Now let \(\Delta (A)=\{\bar x|\) \(x\in eRe\) and xA\(\subset A\}\), and denote by \(N^{(m)}\) the direct sum of m copies of N. There is a mapping \(\theta\) (m) of isomorphism classes of maximal submodules of \(N^{(m)}\) into the isomorphism classes of maximal submodules of \(N^{(m+1)}\). The main result of this paper is a slight generalization of earlier results of the first-named author [cf. \textit{M. Harada}; Part I, II, ibid. 21, 649-670, 671-677 (1984; Zbl 0543.16007, Zbl 0543.16008]. The authors show: Let \(N=eR/A\) be a hollow module. The following conditions are equivalent: 1) \([\Delta:\Delta (a)]=k\), 2) If \(m>k\), every maximal submodule M in \(D=N^{(m)}\) contains a direct summand of D; 3) \(\theta\) (i) is not epic for every \(i\leq k-1\), but \(\theta\) (j) is epic for every \(j\geq k\). Another result deals with \(D=\oplus^{n}_{i=1}N_ i\), where \(N_ i=eR/A_ i\) is a hollow module. A necessary and sufficient condition is given in order that D satisfies the condition, mentioned before, in terms of elements.