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.


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