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?

Question

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?

Answer 1

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