Almost M-projectives and Nakayama rings
This paper continues the deep and very interesting study of modules that decompose into a direct sum of hollow (indecomposable) modules. Recall that a module is hollow if every proper submodule is small. Let \(\{M_{\alpha}\}\) be a set of nonzero modules, let \(M=\oplus M_{\alpha}\), and let \(\pi_{\alpha}\) be the projection of \(M\) onto \(M_{\alpha}\). Then \(M\) satisfies condition \((D_ 1')\) if every submodule \(N\) of \(M\) such that \(\pi_{\alpha}(N)=M_{\alpha}\) for some \(\alpha\) contains a nonzero direct summand of \(M\). For modules \(K\) and \(L\), \(L\) is almost \(K\)-projective if, for any natural epimorphism \(\nu:L\to L/N\) and homomorphism \(h:K\to L/N\), either (1) \(h\) lifts to \(\bar h:K\to L\) or (2) there exists a direct summand \(L'\) of \(L\) and a homomorphism \(h': L'\to K\) with \(hh'=\nu | L'\). Under the hypothesis that each \(M_{\alpha}\) is a hollow module with local endomorphism ring, condition \((D_ 1')\) is compared with the condition: \(M_{\beta}\) is almost \(\oplus_{\alpha \neq \beta}M_{\alpha}\)-projective. Extensive consideration is then given to the case in which the ring \(R\) is semiperfect, each \(M_{\alpha}\) is uniserial, and the number of \(M_{\alpha}'s\) is finite. The relation of this decomposition theory to the study of semiperfect Nakayama rings is elucidated and basic semiperfect Nakayama rings are characterized. These results are deep and technical. Other applications of the theory are also given. For example, if \(R\) is a local Dedekind domain, then every finite direct sum \(N\) of copies of a hollow module satisfies the condition: for any submodule \(K\) of \(N\), there exists a direct decomposition \(N=N_ 1\oplus N_ 2\) such that \(N_ 1\subseteq K\) and \(K\cap N_ 2\) is small in \(N_ 2\).
scientific article; zbMATH DE number 4132386
scientific article