BOUQUETS OF BAER MODULES

In this paper some transcendental numbers are used to construct infini te-dimensional indecomposable Baer modules. Let R be a ring whose cate gory 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 pa th composition. When R is a path algebra obtained from an extended Cox eter-Dynkin diagram with no oriented cycles, we characterize Baer modu les of countable rank. This characterization is used to show that modu les constructed from Liouville sequences yield a family, B = {B(n)}n=0 infinity, of Baer modules satisfying the following conditions: every e xtension Of B(m) by B(n) splits for every pair (m, n); if m not-equal n, B(m) is not isomorphic to B(n), while automorphisms of B(n) are giv en 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 charac terization is the determination of the rank-one modules with the prope rty that every submodule of infinite rank has a nonzero direct summand that is Baer. In analogy with aleph(r)-free modules, we define aleph( r)-Baer modules and give an example of an aleph1-Baer module that is n ot Baer. The existence of a Baer module, M, that is not a direct sum o f Baer modules of countable rank is also proved. However every nonzero submodule of M has a nonzero direct summand. A problem suggested by t hese results is the existence and structure of indecomposable Baer mod ules of uncountable rank.