Finite generation of symmetric ideals

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 SX be the symmetric group of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the group ring R[SX]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX]-modules. The proof involves introducing a certain partial order on monomials and showing that it is a well-quasi-ordering. 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