Let $n\ge2$ be an integer, $\mathcal{ K}_n$ the Weyl algebra over the Laurent polynomial algebra $A_n=\mathbb{ C} [x_1^{ \pm1}, x_2^{ \pm1}, ..., x_n^{ \pm1}]$, and $\mathbb{ S}_n$ the Lie algebra of divergence zero vector fields on an $n$-dimensional torus. For any $\mathfrak{ sl}_n$-module $V$ and any module $P$ over $\mathcal{ K}_n$, we define an $\mathbb{ S}_n$-module structure on the tensor product $P\otimes V$. In this paper, necessary and sufficient conditions for the $\mathbb{ S}_n$-modules $P\otimes V$ to be simple are given, and an isomorphism criterion for nonminuscule $\mathbb{ S}_n$-modules is provided. More precisely, all nonminuscule $\mathbb{ S}_n$-modules are simple, and pairwise nonisomorphic. For minuscule $\mathbb{ S}_n$-modules, minimal and maximal submodules are concretely constructed.
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