Minimal Generators for Symmetric Ideals

Abstract

Let $K$ be a field, and let $R = K[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $K$. Let ${\mathfrak S}_{X}$ be the symmetric group of $X$. The group ${\mathfrak S}_{X}$ acts naturally on $R$, and this in turn gives $R$ the structure of a left module over the (left) group ring $R[{\mathfrak S}_{X}]$. A recent theorem of Aschenbrenner and Hillar states that the module $R$ is Noetherian. We prove that submodules of $R$ can have any number of minimal generators.