Finite Generation of Symmetric Ideals in a Countable Number of Variables

Let A be a commutative Noetherian ring, and let R = A[x1, x2, . . .] be the polynomial ring in an infinite number of variables xi, indexed by the positive integers. Let S∞ be the symmetric group on an infinite number of letters {1, 2, 3, . . .}. The group S∞ gives a natural action on R, and this in turn gives R the structure of a left module over the (left) group ring RS∞. We prove that ideals I ⊆ R invariant under the action of S∞ are finitely generated as RS∞-modules. The proof involves introducing a new partial order on monomials and showing that it is a quasi-well-ordering. We also introduce 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.