Commutative rings and algebras
Commutative rings and algebras are sets like the set of integers, allowing addition and (commutative) multiplication. Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
A commutative ring is a set endowed with two binary operations '+' and '*' subject to familiar associative, commutative, and distributive laws. (It is usually but not universally assumed that the rings contain an identity element '1' for multiplication.) Examples include the rings of integers in algebraic number fields; here, the interest is number-theoretic: common questions concern factorization and the class group, the action of the Galois group, and the structure of the group of units. A commutative algebra is a commutative ring which contains a field (usually as a subring over which the entire ring is finitely-generated). Examples include coordinate rings of algebraic varieties, that is, quotients of polynomial rings over a field; here, the interest is geometric: how are the local rings different at singular points, and how do subvarieties intersect?
In some sense the theory of commutative rings and algebras can be seen as the search for common features of these two classes of examples, and the effort to explain features of a general commutative ring as being like these two types. We can clarify these fields of inquiry by reviewing the subfields of section 13.
In basic commutative ring theory we establish the main consequences of the definitions of rings (here commutative, although many of the key aspects carry over to general associative rings). Central to the subject are the ideals in the ring, that is, the additive subgroups which are invariant under the multiplication by arbitrary ring elements. These are naturally related to the quotients (homomorphic images) of the ring; in particular we can distinguish certain classes of ideals (e.g. the prime ideals) as those whose quotients are particular types of commutative rings (in this case, integral domains).
Most strong results in commutative ring theory must assume some kind of finiteness condition in order to exclude pathologies. The Noetherian condition is usually strong enough and yet applies to most rings of general interest; this is the assumption that increasing chains of ideals terminate. In particular, chains of prime ideals terminate; the Krull dimension of a ring is the maximum length of such a chain. Stronger results may apply to the more restricted class of Artinian rings (decreasing chains also terminate), a condition which is significant among commutative algebras but for example not even satisfied by the ring of integers. Hilbert's Basis Theorem (that R[X] is Noetherian if R is) established the importance of this topic by allowing a proof of the finite generation of rings of invariants under linear group actions.
Ideals may be added, multiplied, or intersected, giving a sort of combinatorial structure to the set of ideals in a ring; in particular, the prime ideals play a special role among the set of ideals in this way. In the case of number rings, this is more or less the setting for restoring the unique decomposition into primes (and indeed this is the origin of the word 'ideal' due to Kummer). In the case of geometric algebras, this is the setting for families of subvarieties, including nested and intersecting ones. Probably the most important result in this area is the Lasker-Noether theorem decomposing ideals as intersections of primary ideals.
Many results are not only more generally applicable but technically easier for study when stated for modules rather than simply for ideals. A module is an abelian group admitting an action by the ring, and thus includes both ideals and quotient rings but is a broader category admitting sub- and quotient objects as well as direct sums and tensor products. There is a large structure theory available (e.g. the Jordan-Hölder decomposition, Krull-Schmidt theorem, classification of irreducible or simple modules) for some classes of rings. Special classes of modules warrant particular attention (cyclic, free, projective, injective, flat, simple, torsion, nilpotent) as points of reference in general structure theorems or for applications to other ring-theoretic questions. For example, a major recent result (Serre's conjecture) is that finite projective modules over a polynomial ring are free (Quillen and Suslin).
Next we can clarify several topics which are perhaps most easily understood by their connection with geometry. Hilbert's Nullstellensatz establishes a bijection between algebraic varieties in affine n-space and (radical) ideals in the polynomial ring C[X_1,,X_n] over, say, the complex field. In order to study the local behaviour of a variety at a point, we study the corresponding local ring. (If P is a prime ideal of R, the localization R_P is the set of fractions of elements in R whose denominators are outside P. This localization is a local ring: it has a unique maximal ideal ideal.) Nonsingular points on the variety correspond to regular local rings, which obviously can be expected to be better behaved (for example, Serre proved these are the local rings of finite global dimension; Auslander-Buchsbaum showed all are unique factorization domains) and which are of wide applicability in commutative algebra. But the local rings at singular points are also of interest, and allow for a good description of geometric concepts such as classifications of singularities and multiplicities. In this direction we find various special classes of rings: complete intersections, Gorenstein, Cohen-Macaulay, and Buchsbaum rings.
Also motivated by modern algebraic geometry is the use of homological methods. Rather than looking at a single module for the ring, we study complexes: sequences of modules and maps between them; the particular choice of complex depends on the application. For example one may refine the concept of generators of a module to a complex of generators, relations among them, syzygies among those relations, and so on. By taking (co)homology of appropriate complexes we obtain cohomology groups which are the natural setting for intersection theory in algebraic geometry.
Heading away from algebras towards rings like the integers, we turn first toward ring extensions. We have already mentioned localization; within such extensions we can identify the integral closure. This is the most appropriate arena in which to pursue generalizations of Galois Theory. Natural questions in algebraic extensions include the splitting or ramification of prime ideals. One can also consider transcendental extensions, e.g. polynomial rings over general commutative rings; naturally the results tend to blend the ring-theoretic description of the base ring with results of commutative algebras over fields as in previous paragraphs. Of course when starting with rings with additional structure -- a grading, topology, or order, say -- it is natural to investigate whether that additional structure likewise extends.
Next we can identify various additional axioms which define rings of arithmetic interest. In most cases a necessary condition for further progress is the assumption that the ring is a domain (ab=0 only if a=0 or b=0). In particular, such a ring has no nilpotent ideal (an ideal I with I^n=0 for some n). (As long as a ring has no nilpotent ideal, many questions about it reduced to a study of integral domains by using the Chinese Remainder Theorem.) Note that a domain may always be embedded into a (minimal) field, its quotient field, so tools from Field Theory may be applied.
Other axioms are added according to the intended application. For example, principal ideal domains (in which every ideal is generated by one element) include the integers, as well as F[X] where F is a field; the classification of finitely-generated modules over these rings thus provides a classification of finitely-generated Abelian groups in the one case, and a classification of normal forms of matrices in the other. These include Euclidean rings as a special case (rings, such as the Gaussian integers, having a 'size' function consistent with a division algorithm). PID's are a special case of Unique Factorization Domains, although in fact for Noetherian rings R, R is a UFD iff its minimal prime ideals are principal. Other examples of rings of arithmetic interest are the Dedekind domains of number theory, Prüfer rings (every finitely generated regular ideal is invertible), Marot rings (every regular ideal is generated by regular elements), valuation rings (akin to p-adic rings), and others.
Other topics of arithmetic and combinatorial interest include the study of finite commutative rings (including such classic results as Wedderburn's theorem: finite integral domains are fields), and the study of Witt rings (related to power-series rings and particularly useful for studying rings in finite characteristic).
Note that local rings, graded rings, and valuation rings may be given a natural topology. As is done with the p-adic numbers, one frequently embeds such a ring into a completion in which it is possible to perform approximate constructions and then take a limit (Hensel's Lemma). Rings may also acquire a topology from an ordering, or from their connection with analysis (e.g. a ring of functions on a complex manifold).
Commutative algebra has been applied to symbolic and computational questions as well. For example, one may study rings of functions on which differentiation is defined (e.g. power-series ring) and ask for a description of which elements have antiderivates within this ring. A topic of much recent issue is the creation of normal forms for ideals in polynomial rings (e.g. Gröbner bases) and algorithms which may be developed for them to determine membership in an ideal, intersection of ideals, and so on.
Commutative rings and algebras
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers Z, and p-adic integers.Commutative algebra is the main technical tool in the local study of schemes.The study of rings which are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
The subject, first known as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert 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 complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noether, to whom we owe much of the abstract and axiomatic approach to the subject. Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker–Noether theorem.
Much of the modern development of commutative algebra emphasizes modules. 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 ring extensions. Though it was already incipient in Kronecker's work, the modern approach to commutative algebra using module theory is usually credited to Emmy Noether.
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.
Definition and first examples
Definition : A ring is a set R equipped with two binary operations, i.e. operations that combine any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by '+' and ' ', e.g. a + b and a b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, such that multiplication distributes over addition, i.e. a (b + c) = (a b) + (a c). The identity elements for addition and multiplication are denoted 0 and 1, respectively.
If, in addition, the multiplication is also commutative, i.e.
a b = b a,
the ring R is called commutative. In the sequel of this article, all rings will be commutative, unless explicitly stated otherwise.
An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen (numbers).
A field is a commutative ring where every non-zero element a is invertible, i.e. has a multiplicative inverse b such that a b = 1. Therefore, by definition, any field is a commutative ring. The rational, real and complex numbers form fields.
The ring of 2-by-2 matrices is not commutative, since matrix multiplication fails to be commutative, as the following example shows:, which is not equal to
However, matrices that can be diagonalized with the same similarity transformation do form a commutative ring. An example is the set of matrices of divided differences with respect to a fixed set of nodes.
If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R[X]. The same holds true for several variables.
If V is some topological space, for example a subset of some Rn, real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold.
Ideals and the spectrum
In contrast to fields, where every element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation. First, an element a of a R is called a unit if it possesses a multiplicative inverse. Another particular type of element are zero divisors, i.e. non-zero elements a such that there exists a non-zero element b of the ring such that ab = 0. If R possesses no zero divisors, it is called an integral domain since it closely resembles the integers in some ways.
Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably.
Ideals and factor rings
The inner structure of a commutative ring is determined by considering its ideals, i.e. nonempty subsets that are closed under multiplication with arbitrary ring elements and addition: for all r in R, i and j in I, both ri and i + j are required to be in I. Given any subset F = j ∈ J of R (where J is some index set), the ideal generated by F is the smallest ideal that contains I. Equivalently, it is given by finite linear combinations
r f + r2f2 + + rnfn.
An ideal generated by one element is called principal ideal. A ring all of whose ideals are principal is called a principal ideal ring, two important cases are Z and k[X], the polynomial ring over a field k. Any ring has two ideals, namely the zero ideal and R, the whole ring. Any ideal that is not contained in any proper ideal (i.e. ≠R) is called maximal. An ideal m is maximal if and only if R / m is a field. Any ring possesses at least one maximal ideal, a statement following from Zorn's lemma, which is equivalent to the axiom of choice.
The definition of ideals is such that 'dividing' I 'out' gives another ring, the factor ring R / I: it is the set of cosets of I together with the operations
(a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I.
For example, the ring Z/nZ (also denoted Zn), where n is an integer, is the ring of integers modulo n. It is the basis of modular arithmetic.
The localization of a ring is the counterpart to factor rings insofar as in a factor ring R / I certain elements (namely the elements of I) become zero, whereas in the localization certain elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if S is a multiplicatively closed subset of R (i.e. whenever s, t ∈ S then so is st) then the localization of R at S, or ring of fractions with denominators in S, usually denoted S−1R consists of symbols
with r ∈ R, s ∈ S
subject to certain rules that mimick the cancellation familiar from rational numbers. Indeed, in this language Q is the localization of Z at all nonzero integers. This construction works for any integral domain R instead of Z. The localization (R )−1R is called the quotient field of R. If S consists of the powers of one fixed element f, the localisation is written Rf.
Prime ideals and the spectrum
A particularly important type of ideals are prime ideals, often denoted p. This notion arose when algebraists (in the 19th century) realized that, unlike in Z, in many rings there is no unique factorization into prime numbers. (Rings where it does hold are called unique factorization domains.) By definition, a prime ideal is a proper ideal such that, whenever the product ab of any two ring elements a and b is in p, at least one of the two elements is already in p. (The opposite conclusion holds for any ideal, by definition). Equivalently, the factor ring R / p is an integral domain. Yet another way of expressing the same is to say that the complement R p is multiplicatively closed. The localisation (R p)−1R is important enough to have its own notation: Rp. This ring has only one maximal ideal, namely pRp. Such rings are called local.
By the above, any maximal ideal is prime. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult.
The spectrum of Z
Prime ideals are the key step in interpreting a ring geometrically, via the spectrum of a ring Spec R: it is the set of all prime ideals of R.[nb 1] As noted above, there is at least one prime ideal, therefore the spectrum is nonempty. If R is a field, the only prime ideal is the zero ideal, therefore the spectrum is just one point. The spectrum of Z, however, contains one point for the zero ideal, and a point for any prime number p (which generates the prime ideal pZ). The spectrum is endowed with a topology called the Zariski topology, which is determined by specifying that subsets D(f) = , where f is any ring element, be open. This topology tends to be different from those encountered in analysis or differential geometry; for example, there will generally be points which are not closed. The closure of the point corresponding to the zero ideal 0 ⊂ Z, for example, is the whole spectrum of Z
The notion of a spectrum is the common basis of commutative algebra and algebraic geometry. Algebraic geometry proceeds by endowing Spec R with a sheaf (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an affine scheme. Given an affine scheme, the underlying ring R can be recovered as the global sections of . Moreover, the established one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any f : R → S gives rise to a continuous map in the opposite direction
Spec S → Spec R, q f−1(q), i.e. any prime ideal of S is mapped to its preimage under f, which is a prime ideal of R.
The spectrum also makes precise the intuition that localisation and factor rings are complementary: the natural maps R → Rf and R → R / fR correspond, after endowing the spectra of the rings in question with their Zariski topology, to complementary open and closed immersions respectively.
Altogether the equivalence of the two said categories is very apt to reflect algebraic properties of rings in a geometrical manner. Affine schemes are–much the same way as manifolds are locally given by open subsets of Rn–local models for schemes, which are the object of study in algebraic geometry. Therefore, many notions that apply to rings and homomorphisms stem from geometric intuition.
As usual in algebra, a function f between two objects that respects the structures of the objects in question is called homomorphism. In the case of rings, a ring homomorphism is a map f : R → S such that
f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1.
These conditions ensure f(0) = 0, but the requirement that the multiplicative identity element 1 is preserved under f would not follow from the two remaining properties. In such a situation S is also called an R-algebra, by understanding that s in S may be multiplied by some r of R, by setting r · s := f(r) · s.
The kernel and image of f are defined by ker (f) = and im (f) = f(R) = . Both kernel and image are subrings of R and S, respectively.
The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules, which are similar to vector spaces, except that the base is not necessarily a field, but can be any ring R. The theory of R-modules is significantly more difficult than linear algebra of vector spaces. Module theory has to grapple with difficulties such as modules not having bases, that the rank of a free module (i.e. the analog of the dimension of vector spaces) may not be well-defined and that submodules of finitely generated modules need not be finitely generated (unless R is Noetherian, see below).
Ideals within a ring R can be characterized as R-modules which are submodules of R. On the one hand, a good understanding of R-modules necessitates enough information about R. Vice versa, however, many techniques in commutative algebra that study the structure of R, by examining its ideals, proceed by studying modules in general.
Noetherian rings : A ring is called Noetherian (in honor of Emmy Noether, who developed this concept) if every ascending chain of ideals
0 ⊆ I0 ⊆ I1 ⊆ In ⊆ In + 1 ⊆
becomes stationary, i.e. becomes constant beyond some index n. Equivalently, any ideal is generated by finitely many elements, or, yet equivalent, submodules of finitely generated modules are finitely generated. A ring is called Artinian (after Emil Artin), if every descending chain of ideals
R ⊇ I0 ⊇ I1 ⊇ In ⊇ In + 1 ⊇
becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, Z is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain Z Z Z Z shows. In fact, every Artinian ring is Noetherian.
Being Noetherian is an extremely important finiteness condition. The condition is preserved under many operations that occur frequently in geometry: if R is Noetherian, then so is the polynomial ring R[X1, X2, , Xn] (by Hilbert's basis theorem), any localization S−1R, factor rings R / I.
The Krull dimension (or simply dimension) dim R of a ring R is a notion to measure the 'size' of a ring, very roughly by the counting independent elements in R. Precisely, it is defined as the supremum of lengths n of chains of prime ideals
0 ⊆ p0 ⊆ p1 ⊆ ⊆ pn.
For example, a field is zero-dimensional, since the only prime ideal is the zero ideal. In fact, a ring is zero-dimensional (i.e. rings all of whose prime ideals are maximal) if and only if it is Artinian. The integers are one-dimensional: any chain of prime ideals is of the form 0 = p0 ⊆ pZ = p1, where p is a prime number since any ideal in Z is principal.The dimension behaves well if the rings in question are Noetherian: the expected equality
dim R[X] = dim R + 1
holds in this case (in general, one has only dim R + 1 ≤ dim R[X] ≤ 2 · dim R + 1). Furthermore, since the dimension depends only on one maximal chain, the dimension of R is the supremum of all dimensions of its localisations Rp, where p is an arbitrary prime ideal. Intuitively, the dimension of R is a local property of the spectrum of R. Therefore, the dimension is often considered for local rings only, also since general Noetherian rings may still be infinite, despite all their localisations being finite-dimensional.
Determining the dimension of, say, k[X1, X2, , Xn] / (f1, f2, , fm), where k is a field and the fi are some polynomials in n variables, is generally not easy. For R Noetherian, the dimension of R / I is, by Krull's principal ideal theorem, at least dim R − n, if I is generated by n elements. If the dimension does drops as much as possible, i.e. dim R / I = dim R - n, the R / I is called a complete intersection.
A local ring R, i.e. one with only one maximal ideal m, is called regular, if the (Krull) dimension of R equals the dimension (as a vector space over the field R / m) of the cotangent space m / m2.
Constructing commutative rings
There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the ring so as to make it more readily understandable. For example, an integral domain that is integrally closed in its field of fractions is called normal. This is a desirable property, for example any normal one-dimensional ring is necessarily regular. Rendering a ring normal is known as normalization.
If I is an ideal in a commutative ring R, the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring. This topology is called the I-adic topology. R can then be completed with respect to this topology. Formally, the I-adic completion is the inverse limit of the rings R/In. For example, if k is a field, k[[X]], the formal power series ring in one variable over k, is the I-adic completion of k[X] where I is the principal ideal generated by X. Analogously, the ring of p-adic integers is the I-adic completion of Z where I is the principal ideal generated by p. Any ring that is isomorphic to its own completion, is called complete.
By Wedderburn's theorem, every finite division ring is commutative, and therefore a finite field. Another condition ensuring commutativity of a ring, due to Jacobson, is the following: for every element r of R there exists an integer n > 1 such that rn = r. If, r2 = r for every r, the ring is called Boolean ring. More general conditions which guarantee commutativity of a ring are also known.
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