## A polynomial ring in infinitely many variables contains ideals which are not finitely generated

Let $R$ be a commutative ring with 1. Prove that $R[x_1,x_2,\ldots]$ contains an ideal which is not finitely generated.

Consider the ideal $I = (x_1,x_2,\ldots)$, and suppose $I = (A)$ is finitely generated. Since $A$ is finite, and every element of $A$ is a finite sum of terms in finitely many variables, there exists an index $k$ such that $x_k$ does not appear as a factor of any term in $A$. Moeroever, $A$ does not contain any elements with nonzero constant term. This implies that any element of $(A)$ which has a term with $x_k$ as a factor necessarily has a term with degree at least 2; in particular, $x_k \notin (A)$, a contradiction. So $I$ is not finitely generated.