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Let $K$ be a field of characteristic $p$, where $p$ is a prime. Consider the ring $K[X]$ of the polynomials in the variables commutative and $X=\{x_1,x_2,... \}$. Is every ideal in $K[X]$ generated by monomials $x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n}$, where each $d_i$ is a power of $p$, finitely generated?

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  • $\begingroup$ $K[X]$ is not a Noetherian ring, therefore there exist at least one ideal that is not finitely generated. Is this ideal generated by monomials where its exponents are powers of $p$ finitely genereted? $\endgroup$
    – riva
    Commented Oct 30, 2020 at 16:45

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Of course not; consider the ideal $(x_1^p,x_2^p,x_3^p,\ldots)$.

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  • $\begingroup$ Again, as you did twice here today, you have repeated a hint from a comment (15 minutes prior) into an answer, without attribution. $\endgroup$ Commented Oct 30, 2020 at 22:10
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    $\begingroup$ @BillDubuque Just no, really. $\endgroup$
    – Servaes
    Commented Oct 30, 2020 at 22:41

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