Rings, Modules, Fields

5 Noetherian rings

Noetherian rings are nice rings as we have already seen, and are quite common. Moreover, they are closed under quite a few operations:

Lemma 5.1. Noetherian rings are closed under localization and quotients.

Proof. This follows from the correspondence of ideals. □

Proposition 5.2. \(R\) is Noetherian iff \(R[[x]]\) is.

Proof. If \(R[[x]]\) is, certainly \(R\) is, as the ideals are a sublattice. For the converse, use the same proof as for \(R[x]\) except with the ideals \(J_n\) being the ideal of \(c\) occurring as first coefficients of power series in the \(x^n\) term □

Before proving the next proposition, here is another characterization of prime ideals. We say that \(a|b\) for ideals if \(a \supset b\).

Lemma 5.3. \(\pp \) is prime iff \(\pp |ab \implies \pp |a\) or \(\pp |b\) iff \(\pp |a\cap b \implies \pp |a\) or \(\pp |b\). In fact, \(\pp |ab \iff \pp |a\cap b\).

Proof. If \(\pp \) is prime, and \(\pp |ab\) or \(\pp |a\cap b\), but \(\pp \nmid a,\pp \nmid b\), choose \(x \in a-\pp ,y \in b-\pp \). Since \(xy \in \pp \), this is a contradiction. Conversely, if either of the latter two properties holds, then applying it to principle ideals shows that \(\pp \) is prime. The last statement follows. □

The following lemma gives an important use of the ideal quotient.

Lemma 5.4. If \((I:a)= (r_\alpha )\) and \(I+(a) = (s_\beta +j_\beta a)\) where \(s_\beta \in I, j_\beta \in J\), then \((r_\alpha a,s_\beta ) = I\).

Proof. By definition it is contained in \(I\). Now write \(c \in I\) as \(\sum c_i(s_i+j_ia)\). Then since \(c-\sum c_is_i \in J\), \(\sum c_i j_i \in (I:a)\). □

Corollary 5.5. \(R\) is Noetherian iff every prime ideal is finitely generated.

Proof. Suppose all prime ideals are finitely generated. Then suppose \(R\) were not Noetherian, and let \(I\) be a maximal counterexample. \(I\) is not prime, so \(ab \in I, a,b \notin I\) for some \(a,b\). Then the ideals \((I:a), I+(a)\) are strictly larger than \(I\) so are finitely generated. By the previous lemma, so is \(I\), a contradiction. □

Another nice property of Noetherian rings is they have only finitely many minimal primes, or prime ideals that are minimal in the poset of primes. Minimal primes exist by Zorn’s Lemma. To make the proof of this more motivated, we can introduce some notions from algebraic geometry. The Jacobson radical is the intersection of all the maximal ideals, and the nilradical is the intersection of all the prime ideals. We say a ring is reduced if the nilradical is trivial. We say that an ideal \(I\) is radical if \(x^k \in I \implies x \in I\). Note that reduced rings are exactly the quotients of rings by radical ideals.

Lemma 5.6. The nilradical consists of exactly the nilpotent elements.

Proof. If \(a\) is nilpotent, \(a^n = 0\) for some \(n\), and by induction, we see that \(a\) is in every prime. Conversely if \(a\) is not nilpotent, we can localize away from \(a\) to get a nontrivial ring. Then a maximal ideal of the localization corresponds to a prime ideal not containing \(a\). □

Corollary 5.7. Radical ideals are exactly the ideals that are the intersection of the prime ideals.

Proof. Look at the quotient and use our alternate characterization of the nilradical. □

Definition 5.8. The spectrum of \(R\) (denoted \(\Spec (R)\)) is the collection of prime ideals of \(R\) with the topology where the closed sets are of the form \(\{\pp \in \Spec |\pp \supset I\}\).

There is more structure that we can put but we will not need it now. Note that the maximal ideals are the closed points of \(\Spec (R)\). There is a correspondence between ideals and subsets of \(\Spec \). Given an ideal \(I\), we can consider \(Z(I)\), the set of primes containing \(I\), and conversely given a subset \(X \subset \Spec \), we can consider \(I(X)\), the intersection of all the primes it contains.

Definition 5.9. A closed subset \(X\) of a topological space is called irreducible if \(X = X_1 \cup X_2\) for \(X_i\) closed, then \(X = X_i\) for some \(i\).

The following lemma is a trivial version of Hilbert’s Nullstellensatz, that is much easier to prove since it is for arbitrary rings.

Lemma 5.10. \(I\) and \(Z\) are an adjunction between subsets of \(\Spec (R)\) and ideals of \(R\) that induces an equivalence between radical ideals and closed subsets of \(\Spec (R)\). In particular the unit is the closure and the counit is the radical of an ideal. Moreover prime ideals correspond to irreducible closed subsets.

Proof. Everything except the last statement follows from the previous corollary. The last statement follows from Lemma 5.3. □

Definition 5.11. A Noetherian space is one satisfying descending chain condition on closed sets.

A Noetherian ring has a Noetherian spectrum.

Proposition 5.12. A ring with Noetherian spectrum has finitely many minimal primes.

Proof. The nilradical is the intersection of all minimal primes, so it corresponds to a closed set that is the union of all minimal primes. Then we can decompose the nilradical into at most double the amount of closed sets until each component is prime. If this process doesn’t end after a finite number of steps, we have an infinite strictly descending sequence of closed sets. □

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