Simple reflexive modules over finite-dimensional algebras

Ringel CM (2021)
Journal of Algebra and Its Applications 20(09): 2150166.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable modules which are reflexive). In addition, we present some properties of simple reflexive modules in general. Marczinzik had motivated his question by providing large classes A of algebras such that any algebra in A which is not self-injective has simple modules which are not reflexive. However, as it turns out, most of these classes have the property that any algebra in A which is not self-injective has simple modules which are not even torsionless.
Stichworte
Reflexive module; torsionless module; self-injective algebra
Erscheinungsjahr
2021
Zeitschriftentitel
Journal of Algebra and Its Applications
Band
20
Ausgabe
09
Art.-Nr.
2150166
ISSN
0219-4988
eISSN
1793-6829
Page URI
https://pub.uni-bielefeld.de/record/2957020

Zitieren

Ringel CM. Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications. 2021;20(09): 2150166.
Ringel, C. M. (2021). Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications, 20(09), 2150166. https://doi.org/10.1142/S0219498821501668
Ringel, C. M. (2021). Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications 20:2150166.
Ringel, C.M., 2021. Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications, 20(09): 2150166.
C.M. Ringel, “Simple reflexive modules over finite-dimensional algebras”, Journal of Algebra and Its Applications, vol. 20, 2021, : 2150166.
Ringel, C.M.: Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications. 20, : 2150166 (2021).
Ringel, Claus Michael. “Simple reflexive modules over finite-dimensional algebras”. Journal of Algebra and Its Applications 20.09 (2021): 2150166.

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