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POINTERS: [Texts] 13: 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 in which every nonzero element is invertible are fields; see the separate page for 12: Field Theory. In some sense the best questions about ring theory -- those having to do with the set of ideals in a ring -- are lost in fields, so the development of the two disciplines is rather different. On the other hand, fields often contain interesting subrings (especially in the number-fields, with a well-defined ring of integers). Conversely, the study of a ring is often focused by the examination of related fields, such as the quotients by each of the maximal ideals, or, in the case of integral domains, by the quotient field.
Questions about the ring of integers are usually classified as 11: Number Theory. So too are number-theoretic-like questions about closely-related rings such as the ring of integers in an algebraic number field (e.g., which are unique-factorization domains?); see in particular 11R: Algebraic Number Theory.
Topics for which commutativity is not a necessary assumption (e.g. chain conditions for ideals) are treated together with more general non-commutative rings in 16: Associative Rings and Algebras. If associativity is not assumed, the question is classified with 17: Nonassociative Rings such as Lie algebras. Separate from the MSC classifications, there is at this site a long FAQ on rings with particular emphasis on the study of division rings over the reals.
For studies of sets of ideals in rings (particularly rings of polynomials) see 14: Algebraic Geometry. In some sense algebraic geometry and commutative algebra are simply two languages for the same ideas. Typically one classifies problems as Algebraic Geometry when stated in terms of points, hypersurfaces, divisors, and other geometric objects, and as Commutative Algebra when stated in terms of ideals and coordinate rings, although in practice techniques from both areas are used in tandem. Since the analytic sets in 32: Several Complex Variables are in practice similar to the varieties in algebraic geometry, many of the same ring-theoretic tools may be applied to that setting as well. (especially section 32B).
Rings associated to group a group G shed light on the structure of G, particular rings of invariants k(V)^G (given a group action on a vector space V), cohomology rings H^*(G,Z), group rings Z[G], and representation rings R(G). Typically this material is classified with 20: Group Theory. There are also groups associated to a ring R, such as the additive and multiplicative groups, the general linear groups GL_n(R), and other algebraic and arithmetic groups. (The matrix groups are also studied in 15: Linear Algebra, especially 15A33). It is the systematic study of these groups which leads to 19: Algebraic K-theory.
Commutative ring structures arise naturally in 18: Homological Algebra and 55: Algebraic Topology. (Actually these rings are typically graded and commutative only in the graded sense, that is, a*b=-b*a when both a and b are of odd degree, but such rings are actually commutative in many special cases, and tools from commutative algebra often carry over unchanged in the general case.) Of course the functors of homological algebra (Tor, Ext, etc.) are used to study modules over rings.
Computational techniques particularly in polynomial rings (e.g. Gröbner basis algorithms) have been used in 68: Computer Science (particularly 68W30: Symbolic algebra) to facilitate simplification and the solving of polynomial equations. When applied to geometric problems such as movable linkages this has proven useful in 52: Convex Geometry and 93: Control Theory.
Applications to 05: Combinatorics include the use of modules whose Hilbert series matches the generating series of some combinatorial sequence (e.g. the work of Stanley).
Most of the papers in commutative rings bearing a classification in 03: Mathematical Logic are in 03C60: Model-theoretic algebra.
For Boolean rings see 06: Ordered algebraic structures, especially section 06E.
One common example of a commutative ring is the ring of polynomials over
a field (say). The algebraic study of general collections of
polynomials is appropriate for this field; the study of individual
polynomials or specific collections usually belongs elsewhere. For
example, one may seek the roots of a polynomial numerically in
65: Numerical Analysis; one may view them as
analytic functions in 26: Real Analysis or
30: Complex Analysis. Specific families of
polynomials, e.g. the Chebyshev polynomials, are treated in
33: Special functions, or
11: Number Theory, or 42: Fourier
Series (and orthogonal functions) as appropriate. The general
algebraic inspection of a single integral polynomial could appropriate occur
in many subfields of number theory (11) or field theory (12). For
example, one may look at the extension field it generates in
11R: Algebraic Number Theory or as a part of
12F: Galois theory.
![[Schematic of subareas and related areas]](../images/13b.gif)
This image slightly hand-edited for clarity.
Browse all (old) classifications for this area at the AMS.
Nearly every beginning graduate text in algebra incorporates commutative ring theory with general ring theory and other topics. Classical texts defining the field include
Many texts in algebraic geometry also devote substantial space to the necessary tools in algebra (e.g. Eisenbud's book)
Other distinctive resources:
On-line textbook [John Beachy]
Online notes for a graduate course in ring theory [Lee Lady]
There is a mailing list COM-ALG; archives are available.
Macaulay is designed to allow computations in commutative rings such as chain complex calculations, resolutions, syzygies, and so on.
The Magma system can handle computations in polynomial rings.
CoCoA -- Computations in Commutative Algebra.
Posso, a package for doing computations with Gröbner Bases. A demo on-line server is available for computations of several sorts in polynomial rings.
Fermat is a computer algebra system for matrix and polynomial computations over the rationals (Q), the integers (Z), and finite fields. The main emphasis is for computations such as Smith Normal Form and GCD of multivariate polynomials. One can form quotient rings of polynomial rings and do computations there. It also performs nice graphics. Versions that run as MPW tools can be downloaded. Free, for Macs and Windows machines.