Poor modules with no proper poor direct summands
\textit{Poor modules} have been introduced by \textit{A. N. Alahmadi} et al. [Glasg. Math. J. 52A, 7--17 (2010; Zbl 1228.16004)], as follows. Let \(R\) be an associative ring with nonzero identity element, and let \(M\) be a unital right \(R\)-module. The module \(M\) is called \textit{poor} if whenever \(M\) is \(N\)-injective for some right \(R\)-module then \(N\) is a semisimple module. In the paper under review, the authors investigate poor modules that have an additional property, they call \textit{pauper}: a module \(M\) is said to be \textit{pauper} if it is poor and has no proper poor direct summand. The ring \(R\) was said to have no right middle class in the paper of Alahmadi et al. [loc. cit.] mentioned above if each right \(R\)-module is either injective or poor. Among others, the authors prove in Sections 3 and 4 the following two results. (1) For a ring \(R\) with no right middle class there exists a pauper right \(R\)-module if and if \(R\) is right Noetherian. (2) A commutative Noetherian ring \(R\) has no indecomposable middle class if and only if \(R\) is the direct product of finitely many fields and at most one ring of composition length 2. Section 4 is devoted to the study of poor and pauper modules over Noetherian rings; for commutative hereditary Noetherian rings the poor modules are fully characterized. Section 5 investigates poor and pauper modules over semi-Artinian rings, while in the last Section 6, abelian groups of torsion-free rank 1 are described.
scientific article; zbMATH DE number 6851766
scientific article