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-C{\'a}rdenas and Salmer{\'o}n. We provide sufficient 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 Kiri\v{c}enko and Kostyukevich. We describe the syzygies of uniserial summands of the radical. We give examples of B\"{a}ckstr\"{o}m orders that are string algebras over discrete valuation rings.