Let be a commutative ring with 1. Prove that contains an ideal which is not finitely generated.
Consider the ideal , and suppose is finitely generated. Since is finite, and every element of is a finite sum of terms in finitely many variables, there exists an index such that does not appear as a factor of any term in . Moeroever, does not contain any elements with nonzero constant term. This implies that any element of which has a term with as a factor necessarily has a term with degree at least 2; in particular, , a contradiction. So is not finitely generated.