Page "Noetherian ring" Paragraph 18

Although every ideal in is simply the set of multiples of a certain integer n, the ideal structure of is slightly more complicated ; there are ideals that may not be expressed as the set of multiples of a given polynomial.

Put differently, is not a principal ideal ring.

However, it is a Noetherian ring.

This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert ; the theorem asserts that if R is any Noetherian ring ( such as, for instance, ), R is also a Noetherian ring.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.