On the Stable Radical of Some Non-Domestic String Algebras

Abstract

We introduce the concept of a prime band in a string algebra Λ and use it to associate to Λ its finite bridge quiver. Then we introduce a new technique of ‘recursive systems’ for showing that a graph map between finite dimensional string modules lies in the stable radical. Further we study two classes of non-domestic string algebras in terms of some connectedness properties of its bridge quiver. ‘Meta-\(\bigcup \)-cyclic’ string algebras constitute the first class that is essentially characterized by the statement that each finite string is a substring of a band. Extending this class we have ‘meta-torsion-free’ string algebras that are characterized by a dichotomy statement for ranks of graph maps between string modules–such maps either have finite rank or are in the stable radical. Their stable ranks can only take values from {ω, ω + 1,ω + 2}.