1

Let $R$ be a ring, and let $M$ be an $R$-module. Can we characterize when the canonical map $M \rightarrow [[M, R]_{R \text{-mod}} , R ]_{R \text{-mod}}$ sending $a$ to $\hat{a} : [M, R]_{R \text{-mod}} \rightarrow R$ sending $\phi$ to $\phi(a)$ is an isomorphism?

If $R$ is a field and $M$ is finite dimensional over $R$, then this map is an isomorphism. Are there other criteria on $R$ and $M$ for this map to be an isomorphism? What is a general context for this to hold?

Bernard
  • 175,478
Ronald J. Zallman
  • 1
  • 5
    I doubt there is a nice characterization. For instance, here is a remarkable theorem: for $R=\mathbb{Z}$, every free module $M$ satisfies $M^{**}\cong M$ iff no measurable cardinals exist. – Eric Wofsey Feb 20 '21 at 23:23
  • @EricWofsey Oh dear, I'd love to have some references. – juan diego rojas Feb 20 '21 at 23:59
  • 1
    Such modules are called reflexive modules. There is a paper by Pierre Samuel (in french), entitled Modules réflexifs et anneaux factoriels (Reflexive modules and U.F.D.s). – Bernard Feb 21 '21 at 00:16
  • A simple observation - far from a general classification - is that this holds for projective and finitely generated $M$ (informally, this may be somewhat analogous to "finite dimensional" over any ring). Here is a proof. – qualcuno Feb 21 '21 at 00:18
  • @juandiegorojas: See https://mathoverflow.net/questions/132073/homomorphisms-from-powers-of-z-to-z – Eric Wofsey Feb 21 '21 at 00:47

0 Answers0