Definition 17.5.1 (Noetherian ring). A commutative ring is Noetherian if it satisfies the ascending chain condition (ACC) on ideals: every ascending chain of ideals eventually stabilizes ( for some ). Equivalently, every ideal of is finitely generated.
Note. Every PID is Noetherian (every ideal is generated by one element). Every field is Noetherian. The ring of integers is Noetherian. However, the polynomial ring in infinitely many variables is not Noetherian: .
Theorem 17.5.1 (Hilbert Basis Theorem). If is a Noetherian ring, then so is the polynomial ring .
Proof. Let be an ideal. For each , let be the set of leading coefficients of polynomials in of degree , together with 0. Each is an ideal of , and . Since is Noetherian, this chain stabilizes at some .
For each , the ideal is finitely generated (since is Noetherian), say by leading coefficients of polynomials of degree . We claim these finitely many polynomials generate .
Given of degree , if , the leading coefficient of lies in , so we can subtract an -linear combination of the (multiplied by appropriate powers of ) to reduce the degree. If , the leading coefficient lies in , and we subtract a combination of the to reduce the degree further. This process terminates, producing as a combination of the chosen generators. ∎
Corollary 17.5.1. For any Noetherian ring and any , the polynomial ring is Noetherian. In particular, is Noetherian for any field .
Theorem 17.5.2 (Gauss’s Lemma and polynomial UFDs). If is a UFD, then so is . In particular, and are UFDs.
Proof (idea). A polynomial is primitive if the gcd of its coefficients is 1. Gauss’s lemma states that the product of primitive polynomials is primitive. The key step: an irreducible element of is either an irreducible of (viewed as a constant polynomial) or a primitive polynomial that is irreducible over the fraction field of . Factoring out content and using unique factorization in and in (a PID, hence UFD) gives the result. ∎