Finite Generation of Symmetric Ideals in a countable number of variables

Let $A$ be a commutative Noetherian ring (for instance, a field), and let $R = A[x_1,x_2,\ldots]$ be the polynomial ring in an infinite number of variables $x_i$, indexed by the positive integers. Let ${\mathfrak S}_{\infty}$ be the symmetric group on an infinite number of letters $\{1,2,3,\ldots\}$. The group ${\mathfrak S}_{\infty}$ gives a natural action on $R$, and this in turn gives $R$ the structure of a left module over the (left) group ring $R{\mathfrak S}_{\infty}$. We prove that ideals $I \subseteq R$ invariant under the action of ${\mathfrak S}_{\infty}$ are finitely generated as $R{\mathfrak S}_{\infty}$-modules. The proof involves introducing a new partial order on monomials and showing that it is a quasi-well-ordering. A motivating question arising from chemistry is also presented. This work is joint with Matthias Aschenbrenner (University of Illinois at Chicago).