  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.
