Modules whose hereditary pretorsion classes are closed under products.
Let \(R\) be a ring with identity. It was shown by \textit{J. A. Beachy} and \textit{W. D. Blair} [Commun. Algebra 6, 1-34 (1978; Zbl 0405.16005)] that the following conditions on a ring \(R\) with identity are equivalent: (1) Every hereditary pretorsion class in \(R\)-Mod is closed under arbitrary direct products. (2) Every left \(R\)-module \(M\) is finitely annihilated, that is, \((0:M)=(0:X)\) for some finite subset \(X\) of \(M\). (3) \(R\) is left Artinian. In this paper, the authors attempt to describe those modules \(M\) with the property that every hereditary pretorsion class in \(\sigma[M]\) is closed under products in \(\sigma[M]\). Such a module \(M\) is called product closed. Then, as is shown by the authors, if \(M\) is finitely generated product closed, and is projective in \(\sigma[M]\), then every hereditary pretorsion class in \(\sigma[M]\) is subgenerated by an \(M\)-generated module. This is one of the authors' main results and extends Beachy and Blair's characterization of left Artinian rings.
scientific article; zbMATH DE number 5124469