Reflexive serial algebras
type: Item
instance of: scholarly article
title: Reflexive serial algebras
zbMATH Open document ID: 0638.16013
DOI: 10.1016/0024-3795(88)90133-4
author: Javad Faghih Habibi
author: William H. Gustafson
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: Let R be a finite-dimensional algebra over a field K and M a left R- module of K-dimension n. Let \(\phi =\phi_ M: R\to M_ n(K)\) be the matrix representation of R afforded by M and A the subalgebra of \(M_ n(K)\) consisting of all linear transformations of M which leave all R- submodules of the K-vector space M invariant. Then \(\phi\) (R)\(\subset A\). If \(\phi (R)=A\), M is called reflexive. The paper is concerned with faithful reflexive R-modules in case R is a Nakayama \((=generalized\) uniserial) algebra and K is algebraically closed. Main result: Let L be the minimal faithful R-module, \(n=number\) of simple R-modules. (i) If L is indecomposable, then \(L\oplus (rad L)\) n is reflexive and is a subquotient module of every reflexive R-module. (ii) If L is not indecomposable, then every faithful R-module is reflexive.
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16Gxx
zbMATH DE Number: 4040113
zbMATH Keywords: generalized uniserial algebra
zbMATH Keywords: finite-dimensional algebra
zbMATH Keywords: matrix representation
zbMATH Keywords: faithful reflexive R-modules
zbMATH Keywords: minimal faithful R-module
zbMATH Keywords: simple R-modules
reviewed by: Herbert Kupisch
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90133-4
OpenAlex ID: W2152271305
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description: scientific article; zbMATH DE number 4040113