Bass orders in non semisimple algebras
type: Item
instance of: scholarly article
title: Bass orders in non semisimple algebras
zbMATH Open document ID: 0826.16011
author: Kenji Nishida
author: Hiroaki Hijikata
published in: Journal of Mathematics of Kyoto University
publication date: 1995-11-13T00:00:00Z
review text: 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 s…
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16L60
zbMATH DE Number: 775701
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: Bass orders
zbMATH Keywords: hereditary orders
zbMATH Keywords: quasi Frobenius algebras
zbMATH Keywords: projective indecomposable left and right modules
zbMATH Keywords: maximal orders
zbMATH Keywords: indecomposable representations
zbMATH Keywords: superGorenstein orders
zbMATH Keywords: nonhereditary Bass orders
zbMATH Keywords: primary Bass orders
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518887
OpenAlex ID: W1603599318
DOI: 10.1215/KJM/1250518887
description: scientific article; zbMATH DE number 775701