On Wild Algebras and Super-Decomposable Pure-Injective Modules

Abstract

Assume that k is an algebraically closed field and A is a finite-dimensional wild k-algebra. Recently, L. Gregory and M. Prest proved that in this case the width of the lattice of all pointed A-modules is undefined. Hence the result of M. Ziegler implies that there exists a super-decomposable pure-injective A-module, if the base field k is countable. Here we give a different and straightforward proof of this fact. Namely, we show that there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed A-modules. This also yields the existence of a super-decomposable pure-injective A-module.