Superdecomposable Complete Path Algebras

Abstract

Generalizing the setting of the second author's work on indecomposable modules without nonzero hollow factors, we characterize quivers $Q$ for which the dual algebra $R$ of the path coalgebra of $Q$, equivalently the complete path algebra, has no nonzero uniform right ideals: this happens exactly when every vertex of $Q$ can reach a vertex emitting at least two distinct arrows. Equivalently, the injective hull $E(R)$ of the right regular module is superdecomposable. These conditions are also equivalent to the path coalgebra having no nonzero hollow factor modules as a left module over its dual algebra.