Hilbert basis theorem

If \(R\) is a commutative Noetherian ring, then the polynomial ring \(R[x]\) is Noetherian. By induction, \(R[x_1, \ldots, x_n]\) is Noetherian for every \(n\); in particular every ideal of \(k[x_1, \ldots, x_n]\) over a field \(k\) is finitely generated, and every ideal of \(\ZZ[x_1,\ldots,x_n]\) is finitely generated.