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On the nonprimality of certain symmetric ideals. (English) Zbl 07823246

Summary: Let \(R=k[x_1, \ldots, x_n, \ldots]\) be the infinite variable polynomial ring equipped with the natural \(\mathfrak{S}_\infty\) action, where \(k\) is a field of characteristic zero. In recent work, R. Nagpal and A. Snowden [“Symmetric ideals of the infinite polynomial ring”, Preprint, arXiv:2107.13027] gave an indirect proof that the \(\mathfrak{S}_\infty\)-ideal generated by \((x_1-x_2)^{2n}\) is not \(\mathfrak{S}_\infty\)-prime. In this paper, we give a direct proof, with explicit elements. We further formulate some conjectures on possible generalizations of the result.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13A50 Actions of groups on commutative rings; invariant theory
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References:

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[5] R. Nagpal and A. Snowden, “Symmetric ideals of the infinite polynomial ring”, preprint, 2021. arXiv: 2107.13027
[6] A. Snowden, “The spectrum of a twisted commutative algebra”, preprint, 2020. arXiv: 2002.01152
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