Let $(R,\mathfrak m)$ be a Noetherian local complete domain of dimension $1$, with fraction field $K$.
(all these assumptions on $R$ imply in particular that for every finitely generated $R$-algebra $S$ lying inside $K$, the integral closure of $S$ in $K$ is a finitely generated $S$-module ) .
For a finitely generated module $M$ over $R$, let the rank of $M$ be $\operatorname {rk} M=\dim_K M\otimes_R K$ . If $M$ is finitely generated and torsion free of rank $r$, then $M\hookrightarrow R^r$, the canonical map $j_M: M\to M^{**}$ is injective and also $M^*=\operatorname{Hom}_R(M,R)$ has same rank as $M$ and is also torsion free. Now for a finitely generated torsion free module $M$, define
$(M:M^{**}):=\{r\in R: rM^{**}\subseteq j_M(M) \}$.
For a fixed $r>0$, let $Tf_r$ be the category of finitely generated torsion-free $R$-modules of rank $r$ and define $T_r:=\cap_{M \in Tf_r } (M: M^{**}) $. So $T_r$ is an ideal of $R$, and my question is:
Is $T_r$ necessarily non-zero ?
Just for clarity, the canonical map $j_M: M\to M^{**}$ is the one that sends $m\in M$ to the map $M^*\to R$ that takes $f\in M^*$ to $f(m)$.