Weak Krull-Schmidt for infinite direct sums of uniserial modules
The purpose of this paper is to determine necessary and sufficient conditions for two arbitrary families \(\{U_i:i\in I\}\) and \(\{V_j:j\in J\}\) of uniserial modules to have the property \(\bigoplus_{i\in I} U_i\cong\bigoplus_{j\in J} V_j\). Most of the results proved by the second author for finite families of uniserial modules extend to arbitrary families [see \textit{A. Facchini}, Trans. Am. Math. Soc. 348, No. 11, 4561-4575 (1996; Zbl 0868.16003)]. The main new idea is that of a quasismall module: a module \(U\) is quasismall if whenever \(U\) is isomorphic to a direct summand of a direct sum \(\bigoplus_{\lambda\in\Lambda}M_\lambda\) of arbitrary modules \(M_\lambda\) (\(\lambda\in\Lambda\)), there is a finite subset \(F\subseteq\Lambda\) such that \(U\) is isomorphic to a direct summand of \(\bigoplus_{\lambda\in F} M_\lambda\). If \(U_i\) (\(i\in I\)) and \(V_j\) (\(j\in J\)) are all quasismall uniserial modules then a complete analogue of Facchini's Theorem is obtained. The class of quasismall uniserial modules includes all finitely generated and all uncountably generated uniserial modules and also all uniserial modules with local endomorphism ring. Another analogue of Facchini's Theorem is obtained in case \(\{U_i:i\in I\}\) and \(\{V_j:j\in J\}\) are locally semi-\(T\)-nilpotent families.
scientific article; zbMATH DE number 1051515
scientific article