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.

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