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Denote by $Q(R)$ the total ring of fractions of $R$, i.e. the localization of $R$ at the regular elements of $R$. Then $m$ torsion depends only on $M := \langle m \rangle$, namely $M \otimes Q(R) = 0$. This can be tested locally on $Spec(R)$ if the canonical maps $Q(R) \to Q(R_{f_i})$ are isomorphisms. So everything is fine if $R$ is an integral domain. This holds also in other examples, but not in general. See this discussion about the sheaf of meromorphic functions.