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Minimal Generators for Symmetric Ideals

Christopher J. Hillar·Troels Windfeldt
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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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Christopher J. Hillar, Troels Windfeldt. 2006-09-27. Minimal Generators for Symmetric Ideals. https://arxiv.org/abs/math/0608003

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