Simple reflexive modules over finite-dimensional algebras

Let [Formula: see text] be a finite-dimensional algebra. If [Formula: see text] is self-injective, then all modules are reflexive. Marczinzik recently has asked whether [Formula: see text] 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 [Formula: see text] of algebras such that any algebra in [Formula: see text] 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 [Formula: see text] which is not self-injective has simple modules which are not even torsionless.

Simple reflexive modules over Artin algebras

Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.