Help proving the ideal generated by $x_1, ...$ is not finitely generated on the ring of polynomial with infinite many variables.

Question

I have problems proving that the ideal $(x_1, x_2, \dots )$ on the ring of polynomials on infinitely many variables $R[x_1, x_2, \dots ]$ is not finitely generated as an ideal. I don't have any idea on how to proceed since the combinations that involve writing $x_i$ as a combination of a set of generators $f_1, \dots, f_m$ i.e. $x_i = g_1 f_1 + g_2 f_2 + \dots g_m f_m$, the $g_i$ can have any variable $x_k$. I thought that it may be helpful to notice that this is a maximal ideal; but I fail to connect this idea to the suppose set of generators $(f_1, \dots, f_m)$.

Answer 1

Hint: let $f_1, \ldots, f_m$ be a finite subset of $R[x_1, x_2, \ldots]$. Then the set of $i$ such that $x_i$ appears in some $f_j$ is finite, and hence has a maximum element, $n$ say. $x_{n+1}$ belongs to the ideal $(x_1, x_2, \ldots)$, but how can it belong to the ideal $(f_1, \ldots, f_m)$?