Steadiness of regular semiartinian rings with primitive factors Artinian.
A module \(M\) is called dually slender if the functor \(\Hom(M,-)\) commutes with direct sums; Hyman Bass observed that this is equivalent to requiring that \(M\) is not the union of a countably infinite increasing chain of proper submodules. A ring \(R\) is called (right) steady if the only dually slender right \(R\)-modules are the finitely-generated ones. The main result of this paper is a ring-theoretic characterization of steady rings for a certain class of rings, the regular semiartinian rings \(R\) such that all primitive factor rings of \(R\) are Artinian. The theorem says that such a ring is steady if and only if no factor ring contains an infinitely generated dually slender (right) ideal, and that the characterization is right-left symmetric. The theorem generalizes one in a paper by the author, \textit{P. Růžička} and \textit{J. Trlifaj} [Lect. Notes Pure Appl. Math. 201, 359-371 (1998; Zbl 0917.16004)].
unions of countably infinite chains of proper submodules
scientific article; zbMATH DE number 5078837