Bass orders in non semisimple algebras
Let \(R\) be a complete discrete valuation domain with quotient field \(K\). For a finite dimensional \(K\)-algebra \(A\), the authors develop a general theory of Bass orders in \(A\) and their representations, which was established for semisimple \(A\) by Drozd, Rojter, and Kirichenko in 1967. In contrast to the classical theory, there is a strict version of Bass order which says that every ``generalized'' overorder (Drozd-Kirichenko) is Gorenstein. It turns out that hereditary orders in \(A\) are necessarily classical, whereas a Bass order in \(A\) is a product of a classical Bass order and a Bass order in a quasi Frobenius algebra \(B\) with \(\text{Rad}^2 B =0\). Such a \(K\)-algebra \(B\) is characterized by the fact that the projective indecomposable left and right modules are of length two. The authors show that Bass orders \(\Lambda\) in \(B\) are just the orders for which the projection into \(B/ \text{Rad }B\) is a maximal order. Moreover, they give an explicit description of such \(\Lambda\) and their indecomposable representations. Their methods are fairly analogous to those used in the classical case. In particular, the rejection lemma (Drozd-Kirichenko) generalizes. A fruitful concept introduced by the authors is that of a superGorenstein order [implicitly used in \textit{Yu. Drozd} and \textit{V. V. Kirichenko}'s paper in Izv. Akad. Nauk SSSR, Ser. Mat. 36, 328-370 (1972; Zbl 0234.13009)], that is, a Gorenstein order \(\Lambda\) such that each indecomposable bijective \(\Lambda\)-representation has isomorphic radical and upper radical. It is proved that ring indecomposable non hereditary Bass orders are superGorenstein and vice versa. As another interesting result, they show that for an \(R\)-order in a quasi Frobenius algebra \(A\), each left ideal is generated by two elements if and only if any overorder is self-dual. Finally, primary Bass orders are classified under some reasonable conditions, and for non-semisimple \(A\), any ring indecomposable strict Bass order is shown to be a full matrix ring over some primary Bass order.
projective indecomposable left and right modules
scientific article; zbMATH DE number 775701