Bouquets of Baer modules
In the category \(R\)-mod over a path \(K\)-algebra over extended Coxeter- Dynkin diagrams with no oriented cycles a torsion theory is introduced in the natural way. A torsion-free \(R\)-module \(M\) of countable rank is Baer (\(\text{Ext}(T,M) = 0\) for each \(T\) torsion) if and only if every finite rank submodule is finite-dimensional if and only if \(M = \bigcup_{n = 1}^ \infty M_ n\), where \(M_ n \subset M_{n+1}\), \(n = 1, \ldots,\) are finite dimensional submodules of \(M\) with \(M/M_ n\) torsion-free (Th. 1.2). The second part investigates systems \(\mathcal B = \{B_ n\}_{n = 0}^ \infty\) of Baer modules which are rigid in the following sense: The automorphism group of \(B_ n\) is the multiplicative group of \(K\), \(B_ n\) and \(B_ m\) are not isomorphic whenever \(m \neq n\) and \(\text{Ext}(B_ m, B_ n) = 0\) for every pair of positive integers \(m\), \(n\) (Th. 2.4). In the last paragraph the \(\aleph_ r\)-Baer modules (every submodule of \(M\) of cardinality less than \(\aleph_ r\) is Baer) are treated. Among other results the existence of an \(\aleph_ 1\)-Baer but not Baer module is proved.
scientific article; zbMATH DE number 584668
scientific article