Special Quasi-triads and Integral Group Rings of Finite Representation Type, I

Abstract

In this paper, we investigate a particular order over a complete discrete valuation ring which we define to be a special quasi-triad. Such orders manifest themselves in several areas, most notably, the study of orders with finitely many non-isomorphic, indecomposable lattices. We are able to classify all the indecomposable lattices over such an order and thereby determine which such special quasi-triads have the property that every indecomposable lattice is isomorphic to a right ideal. This paper is necessary for the classification of lattices over integral group rings of finite type, which is done in another paper by the authors