Bouquets of Baer modules

Abstract

In this paper some transcendental numbers are used to construct infinite-dimensional indecomposable Baer modules. Let R be a ring whose category of modules has a torsion theory. An R-module, M, is Baer if every extension of M by any torsion R-module splits. In this paper, R will be a path algebra, i.e., an algebra whose basis over a field K are the vertices and paths of a directed graph. Multiplication is given by path composition. When R is a path algebra obtained from an extended Coxeter-Dynkin diagram with no oriented cycles, we characterize Baer modules of countable rank. This characterization is used to show that modules constructed from Liouville sequences yield a family, ▪ = { B n } ∞ n=0 , of Baer modules satisfying the following conditions: every extension of B m by B n splits for every pair ( m,n); if m≠ n, B m is not isomorphic to B n , while automorphisms of B n are given by multiplications by nonzero elements of K. Each B n is shown to be a submodule of a rank-one module. Another application of our characterization is the determination of the rank-one modules with the property that every submodule of infinite rank has a nonzero direct summand that is Baer. In analogy with ℵ r -free modules, we define ℵ r -Baer modules and give an example of an ℵ 1-Baer module that is not Baer. The existence of a Baer module, M, that is not a direct sum of Baer modules of countable rank is also proved. However every nonzero submodule of M has a nonzero direct summand. A problem suggested by these results is the existence and structure of indecomposable Baer modules of uncountable rank.