Abstract: The infinite variable polynomial ring R=C[x1,x2,...] is not Noetherian, but it is Noetherian up to the action of the infinite symmetric group, and one may ask which other classical results about Noetherian rings also hold for R up to this symmetry. In this talk, we will define what modules and prime ideals are in this equivariant setting, and discuss some tools (like representations of combinatorial categories) to study the equivariant commutative algebra of R.