String algebras over local rings: Admissibility and biseriality

Abstract

For a path algebra over a noetherian local ground ring, the notion of an admissible ideal was defined by Raggi-Cárdenas and Salmerón. We characterise the conditions for admissibility, and use them to study semiperfect module-finite algebras over local rings whose quotient by the radical is a product of copies of the residue field. We define string algebras over local ground rings and recover the notion introduced by Butler and Ringel when the ground ring is a field. We prove they are biserial in a sense of Kiričenko and Kostyukevich. We describe the syzygies of the uniserial summands of the radical. We give examples of Bäckström orders that are string algebras over discrete valuation rings.