Simple reflexive modules over finite-dimensional algebras

Ringel CM. Simple reflexive modules over finite-dimensional algebras. Journal of Algebra and Its Applications. 2021;20(09): 2150166.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