Recall that a module $ M$ over a ring $ A$ is reflexive in case the canonical evaluation map $ M \rightarrow M^{**}$ is reflexive with $ (-)^{*}$ being the functor $ Hom_A(-,A)$ .

Questions:

1. Can one characterise the local rings such that each simple module is reflexive? (here no assumption on the ring is given other than being local, so this question is a little different than in the other threads)

2. Now assume $ A$ is a local Artin algebra that is not selfinjective. Can the simple $ A$ -module $ S$ be reflexive? I think the answer is no, and one reason (of many) could be that $ S^{**}$ is not indecomposable.

Proof of 2. for commutative rings: We have $ S \cong A/J$ when $ J$ denotes the Jacobson radical. Then $ Hom_A(A/J,A) \cong ann_l(J)$ , the left annihilator of $ J$ . Since $ A$ is commutative this should be isomrphic to the socle of A, which is not simple since $ A$ is not selfinjective. Now we can do the same again to $ S^{*}$ to get that $ S^{**}$ is the direct sum of at least two simple modules.