  Let $A$ be a commutative Noetherian ring, and let $R = A[X]$ be the
polynomial ring in an infinite collection $X$ of indeterminates over $A$. Let
${\mathfrak S}_{X}$ be the group of permutations of $X$. The group ${\mathfrak
S}_{X}$ acts on $R$ in a natural way, and this in turn gives $R$ the structure
of a left module over the left group ring $R[{\mathfrak S}_{X}]$. We prove that
all ideals of $R$ invariant under the action of ${\mathfrak S}_{X}$ are
finitely generated as $R[{\mathfrak S}_{X}]$-modules. The proof involves
introducing a certain well-quasi-ordering on monomials and developing a theory
of Gr\"obner bases and reduction in this setting. We also consider the concept
of an invariant chain of ideals for finite-dimensional polynomial rings and
relate it to the finite generation result mentioned above. Finally, a
motivating question from chemistry is presented, with the above framework
providing a suitable context in which to study it.
