Minimal generators for symmetric ideals

Let R = K[X] be the polynomial ring in in finitely many indeterminates X over a field K, and let G(X) be the symmetric group of X. The group G(X) acts naturally on R, and this in turn gives R the structure of a module over the group ring R[G(X)]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively.