# Rings with only finitely many indecomposable reflexive modules

Let $$R$$ be a ring. Recall that a module $$M$$ is called reflexive in case the natural evaluation map $$M \rightarrow M^{**}$$ (with $$M^{*}=Hom_R(M,R)$$) is an isomorphism. A module is reflexive if and only if its indecomposable summands are reflexive (at least that should be true for noetherian semiperfect rings), and thus it is enough to look at indecomposable reflexive modules for a classification for most rings.

Question: In case there are only finitely many indecomposable reflexive left $$R$$-modules, are there also only finitely many indecomposable reflexive right $$R$$-modules?

(that question can be considered to be two questions, namely one time for general modules and one time only for finitely generated modules).

Additional assumptions on the ring $$R$$ are welcome.

The property of a ring having only finitely many indecomposable reflexive modules seems to be important in various situations, see for example https://www.degruyter.com/view/j/crll.1985.issue-362/crll.1985.362.63/crll.1985.362.63.xml . Is there a modern name for rings having only finitely many indecomposable reflexive modules?

For any ring $$R$$, the functor $$\mbox{Hom}_R(-,R)$$ induces a duality between the categories of left and right reflexive $$R$$-modules (see Corollary 19.40 in Lam's Lectures on Modules and Rings for a more general statement). Since the category of reflexive right (or left) $$R$$-modules is closed under finite direct sums and direct summands, this duality induces a bijection between the isomorphism classes of indecomposable reflexive right $$R$$-modules and the isomorphism classes of indecomposable reflexive left $$R$$-modules. This will also restrict to a bijection for strongly indecomposable reflexive modules (i.e., reflexive modules having local endomorphism rings) since the opposite of a local ring is still local.