Noetherian rings

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:

${\displaystyle I_{1}\subseteq \cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq \cdots }$

there exists an n such that:

${\displaystyle I_{n}=I_{n+1}=\cdots }$

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.)

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

Hilbert's basis theorem

Hilbert's basis theorem has some immediate corollaries:

1. By induction we see that ${\displaystyle R[X_{0},\dotsc ,X_{n-1}]}$ will also be Noetherian.
2. Since any affine variety over ${\displaystyle R^{n}}$ (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal ${\displaystyle {\mathfrak {a}}\subset R[X_{0},\dotsc ,X_{n-1}]}$ and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces.
3. If ${\displaystyle A}$ is a finitely-generated ${\displaystyle R}$-algebra, then we know that ${\displaystyle A\simeq R[X_{0},\dotsc ,X_{n-1}]/{\mathfrak {a}}}$, where ${\displaystyle {\mathfrak {a}}}$ is an ideal. The basis theorem implies that ${\displaystyle {\mathfrak {a}}}$ must be finitely generated, say ${\displaystyle {\mathfrak {a}}=(p_{0},\dotsc ,p_{N-1})}$, i.e. ${\displaystyle A}$ is finitely presented.

Primary decomposition

An ideal Q of a ring is said to be primary if Q is proper and whenever xy ∈ Q, either x ∈ Q or yⁿ ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (p^e) where p is prime and e is a positive integer. Thus, a primary decomposition of (n) corresponds to representing (n) as the intersection of finitely many primary ideals.

The Lasker–Noether theorem, given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

For any primary decomposition of I, the set of all radicals, that is, the set {Rad(Q₁), ..., Rad(Q_t)} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module R/I; that is, the set of all annihilators of R/I (viewed as a module over R) that are prime.

Localization

The localization is a formal way to introduce the "denominators" to a given ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions

${\displaystyle {\frac {m}{s}}}$.

where the denominators s range in a given subset S of R. The archetypal example is the construction of the ring Q of rational numbers from the ring Z of integers.

Completion

A completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have simpler structure than the general ones and Hensel's lemma applies to them.

Zariski topology on prime ideals

The Zariski topology defines a topology on the spectrum of a ring (the set of prime ideals).^[2] In this formulation, the Zariski-closed sets are taken to be the sets

${\displaystyle V(I)=\{P\in \operatorname {Spec} \,(A)\mid I\subseteq P\}}$

where A is a fixed commutative ring and I is an ideal. This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations . To see the connection with the classical picture, note that for any set S of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of V(S) (in the old sense) are exactly the tuples (a₁, ..., a_n) such that (x₁ - a₁, ..., x_n - a_n) contains S; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, V(S) is "the same as" the maximal ideals containing S. Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.