Let $X$ be a topological space. The usual definition of torsion-free sheaf is that sheaf $\mathcal{F}$ of $\mathcal{R}$-modules over $X$ (where $\mathcal{R}$ is a sheaf of rings over $X$) such that $\mathcal{F}(U)$ has no torsion elements.
There are many cases in which a nontrivial sheaf $\mathcal{F}$ could vanish over open sets $\mathcal{F}(U)$, and then, the only element there is zero. How does the definition handle this situation?
In other words: is the trivial ring accepted to be torsion-free in this context?