Let $T=\mathbb{C}[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$ be the ring of Laurent polynomials in $n$ variables. Let $B \subset T$ be a subring such that $T$ is a finitely generated $B$-module. Let $S \subset \mathbb{Z}^n$ be an additively closed subset with $0 \in S$. By the natural embedding $\mathbb{Z}^n \to T$, we consider $S$ to be a subset of $T$. Let $B_S \subset T$ be the subring given by the $B$-span of $S$ in $T$.
What is a necessary and sufficient condition on $S$ so that $B_S$ is integrally closed? I have a particular $B$ in mind, but I suspect there should be a simple answer independent of $B$ that will probably coincide with the special case I care about. For example, maybe if $S$ is a primitive lattice then $B_S$ will be integrally closed.
This seems like it should be a known result but I haven't been able to find it. If there is an analogous result for polynomial rings I'm interested in that as well.
I (mostly) found my answer here: https://www.emis.de/journals/UIAM/PDF/39-59-70.pdf
There is no $B$ here and we're actually looking at the $\mathbb{C}$-span of $S$. It looks like my guess that $S$ needed to be primitive is correct. I'm no longer certain that I actually needed the $B$ I defined in the question so I'm fine with this answer.