Algebras whose projective modules are reflexive
type: Item
instance of: scholarly article
title: Algebras whose projective modules are reflexive
zbMATH Open document ID: 0828.16006
DOI: 10.1016/0022-4049(94)00042-H
author: J. F. Watters
author: Kent R. Fuller
author: W. Keith Nicholson
published in: Journal of Pure and Applied Algebra
publication date: 1995-04-10T00:00:00Z
review text: Let \(R\) and \(\Delta\) be rings with identity, and let \(_R M_\Delta\) be a bimodule. Let \(\text{alglat }{_RM_\Delta}= \{\alpha \in \text{End}({_RM_\Delta})\mid \alpha m \in R m\) for all \(m \in M\}\), which contains the set \(\lambda(R)\) of left multiplications by elements of \(R\). One calls \(_R M_\Delta\) reflexive if \(\text{alglat }{_RM_\Delta}= \lambda(R)\). In this paper, the authors study rings whose projective modules are reflexive, emphasizing the case where \(\Delta\) is a field, and \(R\) is a finite-dimensional \(\Delta\)-algebra. A complete classification of such algebras is not given, but it is shown that certain classes of algebras fall into this category. In previous works, the authors have shown that if a finite-dimensional \(R\) with radical \(J\) has all its projective modules reflexive, then \(R\) is split, and \(eJe = 0\) for every primitive idempotent \(e\). A class of algebras with this property is that of split hereditary algebras, for which the authors have shown previously that all projectives are reflexive. The first main result is that if \(R\) has a basic set \(e_1, \dots, e_n\) of primitive idempotents and \(e_i Je_i = 0\) for all \(i\), then every generator is reflexive. Generalizing a notion of Gordon and Small, the authors define a class of rings called quasi-piecewise domains. If a finite-dimensional algebra over an algebraically closed field is a quasi- piecewise domain, then all its projectives are reflexive. If \(R\) is a monomial algebra (zero-relation algebra), then all projectives are reflexive if and only if \(e_i Je_i = 0\) for a basic set of primitive idempotents. Finally, some further examples are found in the class of binomial algebras over an algebraically closed field of characteristic zero.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 741482
zbMATH Keywords: zero-relation algebras
zbMATH Keywords: projective modules
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: split hereditary algebras
zbMATH Keywords: primitive idempotents
zbMATH Keywords: quasi-piecewise domains
zbMATH Keywords: binomial algebras
reviewed by: William H. Gustafson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 741482