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.

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