Commutative algebra

Commutative algebra is the branch of that studies s, their, and  over such rings. Both and  build on commutative algebra. Prominent examples of commutative rings include s; rings of s, including the ordinary s $$\mathbb{Z}$$; and s.

Commutative algebra is the main technical tool in the local study of.

The study of rings that are not necessarily commutative is known as ; it includes, , and the theory of s.

Overview
Commutative algebra is essentially the study of the rings occurring in and.

In algebraic number theory, the rings of s are s, which constitute therefore an important class of commutative rings. Considerations related to have led to the notion of a. The restriction of s to subrings has led to the notions of s and s as well as the notion of of an extension of valuation rings.

The notion of (in particular the localization with respect to a, the localization consisting in inverting a single element and the ) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the s that have only one. The set of the prime ideals of a commutative ring is naturally equipped with a, the. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of, a generalization of algebraic geometry introduced by.

Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of, , s, s, s and many other notions.

History
The subject, first known as, began with 's work on s, itself based on the earlier work of and. Later, introduced the term ring to generalize the earlier term number ring. Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as and classical. In turn, Hilbert strongly influenced, who recast many earlier results in terms of an , now known as the Noetherian condition. Another important milestone was the work of Hilbert's student, who introduced s and proved the first version of the.

The main figure responsible for the birth of commutative algebra as a mature subject was, who introduced the fundamental notions of and  of a ring, as well as that of s. He established the concept of the  of a ring, first for  before moving on to expand his theory to cover general s and s. To this day,  is widely considered the single most important foundational theorem in commutative algebra. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.

Much of the modern development of commutative algebra emphasizes. Both ideals of a ring R and R-algebras are special cases of R-modules, so module theory encompasses both ideal theory and the theory of. Though it was already incipient in work, the modern approach to commutative algebra using module theory is usually credited to  and.

Noetherian rings
In, more specifically in the area of known as , a Noetherian ring, named after , is a ring in which every non-empty set of s has a maximal element. Equivalently, a ring is Noetherian if it satisfies the on ideals; that is, given any chain:


 * $$I_1\subseteq\cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots$$

there exists an n such that:


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

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

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 s and the over a  are both Noetherian rings, and consequently, such theorems as the, the , and the  hold for them. Furthermore, if a ring is Noetherian, then it satisfies the on s. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the.

Hilbert's basis theorem
"Theorem. If R is a left (resp. right), then the R[X] is also a left (resp. right) Noetherian ring."

Hilbert's basis theorem has some immediate corollaries:


 * 1) By induction we see that $$R[X_0,\dotsc,X_{n-1}]$$ will also be Noetherian.
 * 2) Since any over $$R^n$$ (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal $$\mathfrak a\subset R[X_0, \dotsc, X_{n-1}]$$ 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 s.
 * 3) If $$A$$ is a finitely-generated $$R$$-algebra, then we know that $$A \simeq R[X_0, \dotsc, X_{n-1}] / \mathfrak a$$, where $$\mathfrak a$$ is an ideal. The basis theorem implies that $$\mathfrak a$$ must be finitely generated, say $$\mathfrak a = (p_0,\dotsc, p_{N-1})$$, i.e. $$A$$ is.

Primary decomposition
An ideal Q of a ring is said to be  if Q is 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 (pe) 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 , given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:

Lasker-Noether Theorem. Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct ; that is:


 * $$I=\bigcap_{i=1}^t Q_i$$

with Qi primary for all i and Rad(Qi) ≠ Rad(Qj) for i ≠ j. Furthermore, if:


 * $$I=\bigcap_{i=1}^k P_i$$

is decomposition of I with Rad(Pi) ≠ Rad(Pj) for i ≠ j, and both decompositions of I are irredundant (meaning that no proper subset of either {Q1, ..., Qt} or {P1, ..., Pk} yields an intersection equal to I), t = k and (after possibly renumbering the Qi) Rad(Qi) = Rad(Pi) for all i.

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

Localization
The 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
 * $$\frac{m}{s}$$.

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

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


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

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 that the points of V(S) (in the old sense) are exactly the tuples (a1, ..., an) such that (x1 - a1, ..., xn - an) 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.

Examples
The fundamental example in commutative algebra is the ring of integers $$\mathbb{Z}$$. The existence of primes and the unique factorization theorem laid the foundations for concepts such as s and the.

Other important examples are:
 * s $$R[x_1,...,x_n]$$
 * The s
 * Rings of s.

Connections with algebraic geometry
Commutative algebra (in the form of s and their quotients, used in the definition of ) has always been a part of. However, in the late 1950s, algebraic varieties were subsumed into 's concept of a. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, s, both often called algebraic stacks.