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Let $A$ be a ring. Recall that a module $M$ is called reflexive in case the canonical evaluation map $f_M : M \rightarrow M^{**}$ is an isomorphism. Here $(-)^{*}$ denotes the functor $Hom_A(-,A)$.

In the following I restrict to Artin algebras, but the questions are interesting for more general rings.

Questions:

  1. What are the Artin algebras $A$ such that every simple $A$-module is reflexive?

  2. Is every such algebra selfinjective?

In the answers below you find good evidence that the answer to 2. might be yes.

Being reflexive implies being a 2. syzygy module, so I would guess that 2. is correct.

In https://www.sciencedirect.com/science/article/pii/002186938590198X a positive answer was given in case $A$ has additionally dominant dimension at least two. In fact one can prove that $A$ with dominant dimension at least one and all simple modules reflexive implies that $A$ must be selfinjective: Being reflexive implies being a 2. syzygy module and thus every simple module is a submodule of a projective module, which is equivalent to have dominant dimension at least one for algebras with dominant dimension at least one. But the socle of an indecomposable injective non-projective module has dominant dimension zero. Thus $A$ has to be selfinjective.

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Here a proof for a larger class than algebras with dominant dimension at least one. I think something like this should work in general, noting that reflexive simples are not just 1-syzygies but even 2-syzygies.

Using theorem 0.1. of https://ac.els-cdn.com/S0021869396902124/1-s2.0-S0021869396902124-main.pdf?_tid=10084272-e73a-11e7-9cde-00000aab0f02&acdnat=1513962402_c8d10a01fd68e782d6bc657b6afdc390 , assume $A$ is an algebra with injective invelope $I(A)$ that has projective dimension at most one (which is more general than having dominant dimension at last one) and that every simple $A$-module is reflexive. Then every simple module is in $\Omega^1(mod-A)$ and by theorem 0.1. this subcategory is extension closed and thus $\Omega^1(mod-A)=mod-A$. This implies that $D(A) \in \Omega^1(mod-A)$ and $D(A)$ is projective as a submodule of a projective module. Thus $A$ is selfinjective.

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Here a proof for algebras with finite Gorenstein dimension (which includes all algebras of finite global dimension). This gives good evidence that this might be true in general (even for general rings?). Let $A$ have positive finite Gorenstein dimension $g$. Then every module has Gorenstein projective dimension at most $g$ and $g$ equals the maximal Gorenstein projective dimension of a simple module (since every module has a composition series with simple factors). Now in case every simple module $S$ is reflexive, we have $S \in \Omega^2(mod-A)$ for each simple $S$. Thus there exists a short exact sequence $0 \rightarrow S \rightarrow P \rightarrow R \rightarrow 0$, where $P$ is projective. Denoting by $Gpdim(M)$ the Gorenstein projective dimension of a module $M$; this gives that $Gpdim(S) \leq max(Gpdim(P),Gpdim(R)-1) \leq g-1$ for each simple module $S$. This is a contradiciton. Thus at least one simple module is not reflexive.

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