Symmetric group fixed quotients of polynomial rings

Given a representation of a finite group $G$ over some commutative base ring $\mathbf{k}$, the cofixed space is the largest quotient of the representation on which the group acts trivially. If $G$ acts by $\mathbf{k}$-algebra automorphisms, then the cofixed space is a module over the ring of $G$-invariants. When the order of $G$ is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that $G$ is the symmetric group on $n$ letters acting on a polynomial ring by permuting its variables. When $\mathbf{k}$ has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials. Localizing $\mathbf{k}$ at a prime integer $p$ while letting $n$ vary reveals striking behavior in these ideals. As $n$ grows, the ideals stay stable in a sense, then jump in complexity each time $n$ reaches a multiple of $p$.