Pure-injective modules over path algebras

Abstract

The path algebra, R, over a field K, of a directed graph is the algebra with basis the paths and vertices of the graph, with multiplication given by path composition. In this paper the graphs are either Coxeter-Dynkin diagrams or extended Coxeter-Dynkin diagrams. All modules are unital right R-modules. The pure-injective R-modules. i.e., direct summands of direct products of finite-dimensional R-modules, are investigated in this paper. We show that—like the pure-projective modules—they are characterized by systems of cardinal invariants. Using these invariants we identify the pure-injective modules whose direct summands are direct products of finite-dimensional modules. It is also shown that an R-module is pure-projective and pure- injective if it has only finitely many isomorphism classes of finite-dimensional indecomposable submodules. This is a well-known result when R is the path algebra of a Coxeter-Dynkin diagram. The key lemma in the paper is a straightforward result on finite-dimensional modules. We also use it to show that an R-module always has a pure submodule of countable rank. Several properties of R-modules with no proper nonzero pure submodules are obtained.