Let $M$ be a finitely generated reflexive module over a Noetherian local ring $R$. Recall that for a module $M$, $\dim M :=\dim\mathrm{Supp}(M)$, so if $M$ is finitely generated, then $\dim M =\dim R/\mathrm{ann}_R (M)$.
My question is: When can we say that $\dim M =\dim R$ ? In particular, if $R$ is Cohen-Macaulay then can we say $\dim M=\dim R$ ? What happens if $R$ is Generalized Cohen-Macaulay?
My thoughts: I know that $M$ can be embedded in a finite free module, so if $R$ is an integral domain, then $M$ is torsion free, hence faithful, so we're done. I'm not sure what happens for Cohen-Macaulay rings.
Please help.