Vol. 76, No. 2, 1978

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Rings with quivers that are trees

Kent Ralph Fuller and Joel K. Haack

Vol. 76 (1978), No. 2, 371–379

Associated with each artinian ring R are two diagrams called the left and right quivers of R. We generalize a well-known theorem on hereditary serial rings by proving that if these quivers have no closed paths then R is a factor ring of a certain ring of matrices over a division ring. It follows that the categories of finitely generated left and right R-modules are Morita dual to one another. Applying our theorem and theorems of Gabriel and Dlab and Ringel, we show how to write explicit matrix representations of all hereditary algebras of finite module type.

Mathematical Subject Classification
Primary: 16A48, 16A48
Received: 18 January 1977
Published: 1 June 1978
Kent Ralph Fuller
Joel K. Haack