|
|
| Line 1: |
Line 1: |
| {{Ring theory sidebar}}
| | Name: Maribel Buck<br>My age: 25<br>Country: Italy<br>Town: Bosaro <br>Postal code: 45033<br>Address: Via Galvani 34<br><br>my web site; [http://www.Cineypsicoanalisis.org/director/conferencia-sobre-pelicula-de-fellini/ FIFA 15 Coin Hack] |
| In [[abstract algebra]], '''ring theory''' is the study of [[ring (mathematics)|rings]]—[[algebraic structure]]s in which addition and multiplication are defined and have similar properties to those operations defined for the [[integer]]s. Ring theory studies the structure of rings, their [[representation of an algebra|representations]], or, in different language, [[module (ring theory)|modules]], special classes of rings ([[group ring]]s, [[division ring]]s, [[universal enveloping algebra]]s), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as [[homological algebra|homological properties]] and [[PI ring|polynomial identities]].
| |
| | |
| [[Commutative ring]]s are much better understood than noncommutative ones. [[Algebraic geometry]] and [[algebraic number theory]], which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''[[commutative algebra]]'', a major area of modern mathematics. Because these three fields are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, [[Hilbert's Nullstellensatz]] is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, [[Fermat's last theorem]] is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
| |
| | |
| [[Noncommutative ring]]s are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of [[function (mathematics)|function]]s on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of [[noncommutative geometry]] and with the discovery of [[quantum group]]s. It has led to a better understanding of noncommutative rings, especially noncommutative [[Noetherian ring]]s. {{harv|Goodearl|1989}}
| |
| | |
| For the definitions of a ring and basic concepts and their properties, see [[ring (mathematics)]]. The definitions of terms used throughout ring theory may be found in the [[glossary of ring theory]].
| |
| | |
| ==History==
| |
| Commutative ring theory originated in algebraic number theory,
| |
| algebraic geometry, and invariant theory. Central to the development of these subjects
| |
| were the rings of integers in algebraic number fields and algebraic function fields,
| |
| and the rings of polynomials in two or more variables.
| |
| Noncommutative ring theory began with attempts to extend the complex numbers
| |
| to various [[hypercomplex number]] systems. The genesis of the theories of commutative
| |
| and noncommutative rings dates back to the early 19th century, while their
| |
| maturity was achieved only in the third decade of the 20th century.
| |
| | |
| More precisely, [[William Rowan Hamilton]] put forth the [[quaternion]]s and [[biquaternion]]s; [[James Cockle (lawyer)|James Cockle]] presented [[tessarine]]s and [[coquaternion]]s; and [[William Kingdon Clifford]] was an enthusiast of [[split-biquaternion]]s, which he called ''algebraic motors''. These noncommutative algebras, and the non-associative [[Lie algebra]]s, were studied within [[universal algebra]] before the subject was divided into particular [[mathematical structure]] types. One sign of re-organization was the use of [[direct sum of modules#Direct sum of algebras|direct sums]] to describe algebraic structure.
| |
| | |
| The various hypercomplex numbers were identified with [[matrix ring]]s by [[Joseph Wedderburn]] (1908) and [[Emil Artin]] (1928). Wedderburn's structure theorems were formulated for finite-dimensional [[algebra over a field|algebras over a field]] while Artin generalized them to [[Artinian ring]]s.
| |
| | |
| In 1920, [[Emmy Noether]], in collaboration with W. Schmeidler, published a paper about the [[ideal theory|theory of ideals]] in which they defined [[Ideal (ring theory)|left and right ideals]] in a [[ring (mathematics)|ring]]. The following year she published a landmark paper called ''Idealtheorie in Ringbereichen'', analyzing [[ascending chain condition]]s with regard to (mathematical) ideals. Noted algebraist [[Irving Kaplansky]] called this work "revolutionary";{{Sfn |Kimberling|1981|p=18}} the publication gave rise to the term "[[Noetherian ring]]", and several other mathematical objects being called ''[[Noetherian (disambiguation)|Noetherian]]''.{{Sfn |Kimberling|1981|p=18}}{{Sfn |Dick|1981|pp= 44–45}}{{Sfn |Osen|1974|pp=145–46}}
| |
| | |
| ==Commutative rings==
| |
| {{Main|Commutative algebra}}
| |
| A ring is called ''commutative'' if its multiplication is [[commutative]]. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to recover properties known from the [[integer]]s. Commutative rings are also important in [[algebraic geometry]]. In commutative ring theory, numbers are often replaced by [[ideal (ring theory)|ideals]], and the definition of the [[prime ideal]] tries to capture the essence of [[prime number]]s. [[Integral domain]]s, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. [[Principal ideal domain]]s are integral domains in which every ideal can be generated by a single element, another property shared by the integers. [[Euclidean domain]]s are integral domains in which the [[greatest common divisor|Euclidean algorithm]] can be carried out. Important examples of commutative rings can be constructed as rings of [[polynomial]]s and their factor rings. Summary: [[Euclidean domain]] => [[principal ideal domain]] => [[unique factorization domain]] => [[integral domain]] => [[Commutative ring]].
| |
| | |
| ===Algebraic geometry===
| |
| {{Main|Algebraic geometry}}
| |
| [[Algebraic geometry]] is in many ways the mirror image of commutative algebra. A [[scheme (mathematics)|scheme]] is built up out of rings in some sense. [[Alexander Grothendieck]] gave the decisive definitions of the objects used in algebraic geometry. He defined the [[spectrum of a ring|spectrum]] of a commutative ring as the space of prime ideals with [[Zariski topology]], but augments it with a [[sheaf (mathematics)|sheaf]] of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of "polynomial functions" defined on that set. These objects are the "affine schemes"; a general scheme is then obtained by "gluing together" several such affine schemes, in analogy to the fact that general varieties can be obtained by gluing together affine varieties.
| |
| | |
| ==Noncommutative rings==
| |
| {{Main|Noncommutative algebraic geometry|non-commutative algebra}}
| |
| Noncommutative rings resemble rings of [[matrix (mathematics)|matrices]] in many respects. Following the model of [[algebraic geometry]], attempts have been made recently at defining [[noncommutative geometry]] based on noncommutative rings.
| |
| Noncommutative rings and [[associative algebra]]s (rings that are also [[vector space]]s) are often studied via their [[Category theory|categories]] of modules. A [[module (mathematics)|module]] over a ring is an Abelian [[group (mathematics)|group]] that the ring acts on as a ring of [[endomorphism]]s, very much akin to the way [[field (mathematics)|field]]s (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square [[matrix (mathematics)|matrices]] or more generally by rings of endomorphisms of Abelian groups or modules, and by [[monoid ring]]s.
| |
| | |
| ==Some useful theorems==
| |
| General:
| |
| *[[Isomorphism_theorem#Rings|Isomorphism theorems for rings]]
| |
| *[[Nakayama's lemma]]
| |
| | |
| Structure theorems:
| |
| *The [[Artin–Wedderburn theorem]] determines the structure of [[semisimple ring]]s.
| |
| *The [[Jacobson density theorem]] determines the structure of [[primitive ring]]s.
| |
| *[[Goldie's theorem]] determines the structure of [[semiprime ideal|semiprime]] [[Goldie ring]]s.
| |
| *The [[Zariski–Samuel theorem]] determines the structure of a commutative [[principal ideal ring]]s.
| |
| *The [[Hopkins–Levitzki theorem]] gives necessary and sufficient conditions for a [[Noetherian ring]] to be an [[Artinian ring]].
| |
| *[[Morita theory]] consists of theorems determining when two rings have "equivalent" module categories.
| |
| *[[Wedderburn's little theorem]] states that finite [[domain (ring theory)|domains]] are [[field (mathematics)|fields]].
| |
| | |
| == Structures and invariants of rings ==
| |
| | |
| ===Dimension of a commutative ring===
| |
| {{main|Dimension theory (algebra)}}
| |
| | |
| The [[Krull dimension]] of a commutative ring ''R'' is the supremum of the lengths ''n'' of all the increasing chains of prime ideals <math>\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n</math>. For example, the polynomial ring <math>k[t_1, \cdots, t_n]</math> over a field ''k'' has dimension ''n''. The fundamental theorem in the dimension theory states the following numbers coincide for a noetherian local ring <math>(R, \mathfrak{m})</math>:<ref>{{harvnb|Matsumura|1980|loc=Theorem 13.4}}</ref>
| |
| *The Krull dimension of ''R''.
| |
| *The minimum number of the generators of the <math>\mathfrak{m}</math>-primary ideals.
| |
| *The dimension of the graded ring <math>\operatorname{gr}_{\mathfrak{m}}(R) = \oplus_{k \ge 0} \mathfrak{m}^k/{\mathfrak{m}^{k+1}}</math> (equivalently, one plus the degree of its [[Hilbert polynomial]]).
| |
| | |
| A commutative ring ''R'' is said to be [[Catenary ring|catenary]] if any pair of prime ideals <math>\mathfrak{p} \subset \mathfrak{p}'</math> can be extended to a chain of prime ideals <math>\mathfrak{p} = \mathfrak{p}_0 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}'</math> of same finite length such that there is no prime ideal that is strictly contained in two consecutive terms. Practically all noetherian rings that appear in application are catenary. If <math>(R, \mathfrak{m})</math> is a catenary local integral domain, then, by definition,
| |
| :<math>\operatorname{dim}R = \operatorname{ht}\mathfrak{p} + \operatorname{dim}R/\mathfrak{p}</math> | |
| where <math>\operatorname{ht}\mathfrak{p} = \operatorname{dim}R_{\mathfrak{p}}</math> is the [[Height (ring theory)|height]] of <math>\mathfrak{p}</math>. It is a deep [[theorem of Ratliff]] that the converse is also true.<ref>{{harvnb|Matsumura|1980|loc=Theorem 31.4}}</ref>
| |
| | |
| If ''R'' is an integral domain that is a finitely generated ''k''-algebra, then its dimension is the [[transcendence degree]] of its field of fractions over ''k''. If ''S'' is an [[integral extension]] of a commutative ring ''R'', then ''S'' and ''R'' have the same dimension.
| |
| | |
| Closely related concepts are those of [[depth (ring theory)|depth]] and [[global dimension]]. In general, if ''R'' is a noetherian local ring, then the depth of ''R'' is less than or equal to the dimension of ''R''. When the equality holds, ''R'' is called a [[Cohen–Macaulay ring]]. A [[regular local ring]] is an example of a Cohen–Macaulay ring. It is a theorem of Serre that ''R'' is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of ''R''. The significance of this is that a global dimension is a [[homological algebra|homological]] notion.
| |
| | |
| ===Morita equivalence===
| |
| {{main|Morita equivalence}}
| |
| Two rings ''R'', ''S'' are said to be [[Morita equivalent]] if the category of left modules over ''R'' is equivalent to the category of left modules over ''S''. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the [[category theory|category]] of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.
| |
| | |
| === Finitely generated projective module over a ring and Picard group ===
| |
| Let ''R'' be a commutative ring and <math>\mathbf{P}(R)</math> the set of isomorphism classes of finitely generated [[projective module]]s over ''R''; let also <math>\mathbf{P}_n(R)</math> subsets consisting of those with constant rank ''n''. (The rank of a module ''M'' is the continuous function <math>\operatorname{Spec}R \to \mathbb{Z}, \, \mathfrak{p} \mapsto \dim M \otimes_R k(\mathfrak{p})</math>.<ref>{{harvnb|Weibel|loc=Ch I, Definition 2.2.3}}</ref>) <math>\mathbf{P}_1(R)</math> is usually denoted by Pic(''R''). It is an abelian group called the [[Picard group]] of ''R''.<ref>{{harvnb|Weibel|loc=Definition preceding Proposition 3.2 in Ch I}}</ref> If ''R'' is an integral domain with the field of fractions ''F'' of ''R'', then there is an exact sequence of groups:<ref>{{harvnb|Weibel|loc=Ch I, Proposition 3.5}}</ref>
| |
| :<math>1 \to R^* \to F^* \overset{f \mapsto fR}\to \operatorname{Cart}(R) \to \operatorname{Pic}(R) \to 1</math> | |
| where <math>\operatorname{Cart}(R)</math> is the set of [[fractional ideal]]s of ''R''. If ''R'' is a [[Regular ring|regular]] domain (i.e., regular at any prime ideal), then Pic(R) is precisely the [[divisor class group]] of ''R''.<ref>{{harvnb|Weibel|loc=Ch I, Corollary 3.8.1}}</ref>
| |
| | |
| For example, if ''R'' is a principal ideal domain, then Pic(''R'') vanishes. In algebraic number theory, ''R'' will be taken to be the [[ring of integers]], which is Dedekind and thus regular. It follows that Pic(''R'') is a finite group ([[finiteness of class number]]) that measures the deviation of the ring of integers from being a PID.<!-- discuss coordinate ring -->
| |
| | |
| One can also consider the [[group completion]] of <math>\mathbf{P}(R)</math>; this results in a commutative ring K<sub>0</sub>(R). Note that K<sub>0</sub>(R) = K<sub>0</sub>(S) if two commutative rings ''R'', ''S'' are Morita equivalent.
| |
| | |
| {{See also|Algebraic K-theory}}
| |
| | |
| ===Structure of noncommutative rings===
| |
| {{main|Noncommutative ring}}
| |
| | |
| The structure of a [[noncommutative ring]] is more complicated than that of a commutative ring. For example, there exist rings which contain non-trivial proper left or right ideals, but are still [[Simple ring|simple]]; that is contain no non-trivial proper (two-sided). Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the [[nilradical of a ring|nilradical]] of a ring, the set of all nilpotent elements, need not be an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all ''n'' x ''n'' matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.
| |
| | |
| The concept of the [[Jacobson radical]] of a ring; that is, the intersection of all right/left [[Annihilator (ring theory)|annihilators]] of [[Simple module|simple]] right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative.
| |
| | |
| Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of ''n''-by-''n'' [[Matrix (mathematics)|matrices over a field]] is noncommutative despite its natural occurrence in [[geometry]], [[physics]] and many parts of mathematics. More generally, [[endomorphism ring]]s of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the [[Klein four-group]].
| |
| | |
| One of the best known noncommutative rings is the division ring of [[quaternions]].
| |
| | |
| ==Applications ==
| |
| | |
| ===The ring of integers of a number field===
| |
| {{main|Ring of integers}}
| |
| | |
| ===The coordinate ring of an algebraic variety===
| |
| If ''X'' is an [[affine algebraic variety]], then the set of all regular functions on ''X'' forms a ring called the [[coordinate ring]] of ''X''. For a [[projective variety]], there is an analogus ring called the [[homogeneous coordinate ring]]. Those rings are essentially the same things as varieties: they correspond in essentially a unique way. This may be seen via either [[Hilbert's Nullstellensatz]] or scheme-theoretic constructions (i.e., Spec and Proj).
| |
| | |
| === Ring of invariants ===
| |
| A basic (and perhaps the most fundamental) question in the classical [[invariant theory]] is to find and study polynomials in the polynomial ring <math>k[V]</math> that are invariant under the action of a finite group (or more generally reductive) ''G'' on ''V''. The main example is the [[ring of symmetric functions|ring of symmetric polynomials]]: [[symmetric polynomial]]s are polynomials that are invariant under permutation of variable. The [[fundamental theorem of symmetric polynomials]] states that this ring is <math>R[\sigma_1, \ldots, \sigma_n]</math> where <math>\sigma_i</math> are elementary symmetric polynomials.<ref>{{harvnb|Springer|1970|loc=Theorem 1.5.7}}</ref>
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| ==References==
| |
| *[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Ring_theory.html History of ring theory at the MacTutor Archive]
| |
| * {{cite book | author=R.B.J.T. Allenby | title=Rings, Fields and Groups|publisher= Butterworth-Heinemann | year=1991 | isbn=0-340-54440-6}}
| |
| * [[Michael Atiyah|Atiyah M. F.]], [[Ian G. Macdonald|Macdonald, I. G.]], ''Introduction to commutative algebra''. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.
| |
| * {{cite book | author=T.S. Blyth and E.F. Robertson| title=Groups, rings and fields: Algebra through practice, Book 3| publisher= Cambridge university Press| year=1985| isbn=0-521-27288-2}}
| |
| * Faith, Carl, ''Rings and things and a fine array of twentieth century associative algebra''. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0-8218-0993-8
| |
| * Goodearl, K. R., Warfield, R. B., Jr., ''An introduction to noncommutative Noetherian rings''. London Mathematical Society Student Texts, 16. Cambridge University Press, Cambridge, 1989. xviii+303 pp. ISBN 0-521-36086-2
| |
| * Herstein, I. N., ''Noncommutative rings''. Reprint of the 1968 original. With an afterword by Lance W. Small. Carus Mathematical Monographs, 15. Mathematical Association of America, Washington, DC, 1994. xii+202 pp. ISBN 0-88385-015-X
| |
| * [[Nathan Jacobson]], ''Structure of rings''. American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964 ix+299 pp.
| |
| * [[Nathan Jacobson]], ''The Theory of Rings''. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York, 1943. vi+150 pp.
| |
| * {{Cite web | last1=Judson | first1=Thomas W. | title=Abstract Algebra: Theory and Applications | year=1997 | url=http://abstract.ups.edu }} An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source [[GNU Free Documentation License|GFDL]] license.
| |
| * Lam, T. Y., ''A first course in noncommutative rings''. Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001. xx+385 pp. ISBN 0-387-95183-0
| |
| * Lam, T. Y., ''Exercises in classical ring theory''. Second edition. Problem Books in Mathematics. Springer-Verlag, New York, 2003. xx+359 pp. ISBN 0-387-00500-5
| |
| * Lam, T. Y., ''Lectures on modules and rings''. Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999. xxiv+557 pp. ISBN 0-387-98428-3
| |
| * McConnell, J. C.; Robson, J. C. ''Noncommutative Noetherian rings''. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001. xx+636 pp. ISBN 0-8218-2169-5
| |
| * Pierce, Richard S., ''Associative algebras''. Graduate Texts in Mathematics, 88. Studies in the History of Modern Science, 9. Springer-Verlag, New York-Berlin, 1982. xii+436 pp. ISBN 0-387-90693-2
| |
| * Rowen, Louis H., ''Ring theory''. Vol. I, II. Pure and Applied Mathematics, 127, 128. Academic Press, Inc., Boston, MA, 1988. ISBN 0-12-599841-4, ISBN 0-12-599842-2
| |
| * {{Citation | last=Springer | first = Tonny A. | title = Invariant theory | year= 1977| publisher=Springer-Verlag | series=Lecture Notes in Mathematics | volume=585}}
| |
| *{{citation |first=Charles |last=Weibel |url=http://www.math.rutgers.edu/~weibel/Kbook.html |title=The K-book: An introduction to algebraic K-theory }}
| |
| * Connell, Edwin, Free Online Textbook, http://www.math.miami.edu/~ec/book/
| |
| | |
| {{DEFAULTSORT:Ring Theory}}
| |
| [[Category:Ring theory|*]]
| |
| | |
| [[de:Ringtheorie]]
| |
| [[ka:რგოლი (მათემატიკა)]]
| |
| [[ro:Inel (algebră)]]
| |