Fully rigid systems of modules
Let R be a commutative ring and A be an R-algebra. The basic notion of this paper is the definition of a fully rigid system for A on a set I which is a family \(G_ X\) (X\(\subseteq I)\) of faithful right A-modules indexed by the subsets X of I such that, for X,Y\(\subseteq I\), \[ G_ X\subseteq G_ Y\quad (X\subseteq Y),\quad Hom_ R(G_ X,G_ Y)=A\quad (X\subseteq Y),\quad Hom_ R(G_ X,G_ Y)=0\quad (X\varsubsetneq Y). \] The author gives a proof (based on Shelah's arguments from 1974) of the following very useful Theorem: Suppose that A admits a fully rigid system \(H_ X\) (X\(\subseteq I)\) on a set I with \(| I| \geq 5\). Then, for every infinite cardinal \(\lambda \geq | H_ I|\), A admits a fully rigid system \(G_ X\) (X\(\subseteq \lambda)\) with \(| G_ X| =\lambda\) (X\(\subseteq \lambda).\) Remark of the reviewer: We are able to sharpen this result, replacing 5 by 4, see \textit{R. Göbel}, \textit{W. May} [J. Pure Appl. Algebra 65, 29-43 (1990; Zbl 0716.16015)] and \textit{C. Böttinger}, \textit{R. Göbel} [to appear in J. Pure Appl. Algebra (1991)]. Moreover, 4 cannot be replaced by 3, as follows from a result of Simson and Ringel-Tachikawa from 1974 (1975).
scientific article; zbMATH DE number 4171146
scientific article