  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 of algebras such that any
algebra in the class 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 the class which is not self-injective has simple
modules which are not even torsionless.
