Direct sums and direct products of finite-dimensional modules over path algebras

AbstractThe algebras in this paper are over the associative algebras R obtained from extended Coxeter-Dynkin quivers with no oriented cycles. The finite-dimensional indecomposable R-modules can, in principle, be described. Taking direct products and direct sums, respectively, of finite-dimensional R-modules over an infinite indexing set are two natural ways of getting infinite-dimensional R-modules. The latter are the infinite-dimensional pure-projective modules and direct summands of the former are the pure-injective modules. The focus in this paper is on these two classes of infinite-dimensional modules. Every module is a submodule of a pure-injective module and a quotient of a pure-projective module. When is an extension of a pure-injective module by a pure-injective module pure-injective? This question is answered in this paper. The answer is analogous to the answer of the corresponding question for pure-projective modules. The structures of some quotients of direct products of finite-dimensional modules are also obtained