ring theory

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In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
, ring theory is the study of
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery) A ring is a round band, usually of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s in which addition and multiplication are defined and have similar properties to those operations defined for the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s. Ring theory studies the structure of rings, their
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It co ...
, or, in different language,
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...
, special classes of rings (
group ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s,
division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
s,
universal enveloping algebra In mathematics, a universal enveloping algebra is the most general (unital algebra, unital, associative algebra, associative) algebra that contains all representation of a Lie algebra, representations of a Lie algebra. Universal enveloping algebras ...
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 properties and polynomial identities.
Commutative ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
s are much better understood than noncommutative ones.
Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

and
algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
, 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 Commutative algebra is the branch of algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...
'', a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example,
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''wikt:Satz, Satz'') is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of alg ...

is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly,
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...
is stated in terms of elementary
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
Noncommutative ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of
noncommutative geometryNoncommutative geometry (NCG) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
and with the discovery of
quantum group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. It has led to a better understanding of noncommutative rings, especially noncommutative
Noetherian ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s. For the definitions of a ring and basic concepts and their properties, see ''
Ring (mathematics) In mathematics, rings are algebraic structures that generalize field (mathematics), fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a Set (mathematics), set equipped with t ...
''. The definitions of terms used throughout ring theory may be found in ''
Glossary of ring theory Ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies th ...
''.

# Commutative rings

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 formalize properties of the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
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 numbers. Integral domains, 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 domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains 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 polynomials and their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring.

## Algebraic geometry

Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

is in many ways the mirror image of commutative algebra. This correspondence started with
Hilbert's Nullstellensatz Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''wikt:Satz, Satz'') is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of alg ...

that establishes a one-to-one correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring. This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings. Alexander Grothendieck completed this by introducing scheme (mathematics), schemes, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the spectrum of a ring, spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology, and augmented with a sheaf (mathematics), sheaf of rings. These objects are the "affine schemes" (generalization of affine varieties), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a manifold by gluing together the chart (topology), charts of an atlas (topology), atlas.

# Noncommutative rings

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 geometryNoncommutative geometry (NCG) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
based on noncommutative rings. Noncommutative rings and associative algebras (rings that are also vector spaces) 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 endomorphisms, very much akin to the way field (mathematics), fields (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 rings.

## Representation theory

Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebra, abstract
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
ic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

# Some relevant theorems

General *Isomorphism theorem#Rings, Isomorphism theorems for rings *Nakayama's lemma Structure theorems *The Artin–Wedderburn theorem determines the structure of semisimple rings *The Jacobson density theorem determines the structure of primitive rings *Goldie's theorem determines the structure of semiprime ideal, semiprime Goldie rings *The Zariski–Samuel theorem determines the structure of a commutative principal ideal ring *The Hopkins–Levitzki theorem gives necessary and sufficient conditions for a
Noetherian ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
to be an Artinian ring *Morita theory consists of theorems determining when two rings have "equivalent" module categories *Cartan–Brauer–Hua theorem gives insight on the structure of
division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
s *Wedderburn's little theorem states that finite domain (ring theory), domains are field (mathematics), fields Other *The Skolem–Noether theorem characterizes the automorphisms of simple rings

# Structures and invariants of rings

## Dimension of a commutative ring

In this section, ''R'' denotes a commutative ring. The Krull dimension of ''R'' is the supremum of the lengths ''n'' of all the chains of prime ideals $\mathfrak_0 \subsetneq \mathfrak_1 \subsetneq \cdots \subsetneq \mathfrak_n$. It turns out that the polynomial ring $k\left[t_1, \cdots, t_n\right]$ over a field ''k'' has dimension ''n''. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring $\left(R, \mathfrak\right)$: *The Krull dimension of ''R''. *The minimum number of the generators of the $\mathfrak$-primary ideals. *The dimension of the graded ring $\textstyle \operatorname_\left(R\right) = \bigoplus_ \mathfrak^k/$ (equivalently, 1 plus the degree of its Hilbert polynomial). A commutative ring ''R'' is said to be Catenary ring, catenary if for every pair of prime ideals $\mathfrak \subset \mathfrak\text{'}$, there exists a finite chain of prime ideals $\mathfrak = \mathfrak_0 \subsetneq \cdots \subsetneq \mathfrak_n = \mathfrak\text{'}$ that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between $\mathfrak$ and $\mathfrak\text{'}$ have the same length. Practically all noetherian rings that appear in applications are catenary. Ratliff proved that a noetherian local integral domain ''R'' is catenary if and only if for every prime ideal $\mathfrak$, :$\operatornameR = \operatorname\mathfrak + \operatornameR/\mathfrak$ where $\operatorname\mathfrak$ is the Height (ring theory), height of $\mathfrak$. 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

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 $\mathbf\left(R\right)$ the set of isomorphism classes of finitely generated projective modules over ''R''; let also $\mathbf_n\left(R\right)$ subsets consisting of those with constant rank ''n''. (The rank of a module ''M'' is the continuous function $\operatornameR \to \mathbb, \, \mathfrak \mapsto \dim M \otimes_R k\left(\mathfrak\right)$.) $\mathbf_1\left(R\right)$ is usually denoted by Pic(''R''). It is an abelian group called the Picard group of ''R''. If ''R'' is an integral domain with the field of fractions ''F'' of ''R'', then there is an exact sequence of groups: :$1 \to R^* \to F^* \overset\to \operatorname\left(R\right) \to \operatorname\left(R\right) \to 1$ where $\operatorname\left(R\right)$ is the set of fractional ideals 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''. 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. One can also consider the group completion of $\mathbf\left(R\right)$; this results in a commutative ring K0(R). Note that K0(R) = K0(S) if two commutative rings ''R'', ''S'' are Morita equivalent.

## Structure of noncommutative rings

The structure of a noncommutative ring is more complicated than that of a commutative ring. For example, there exist Simple ring, simple rings, containing no non-trivial proper (two-sided) ideals, which contain non-trivial proper left or right ideals. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the 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 rings 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 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 analogous 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 Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see ''wikt:Satz, Satz'') is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of alg ...

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 $k\left[V\right]$ 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 polynomials are polynomials that are invariant under permutation of variable. The fundamental theorem of symmetric polynomials states that this ring is $R\left[\sigma_1, \ldots, \sigma_n\right]$ where $\sigma_i$ are elementary symmetric polynomials.

# 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 quaternions and biquaternions; James Cockle (lawyer), James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called ''algebraic motors''. These noncommutative algebras, and the non-associative Lie algebras, 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 rings 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 rings. 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 conditions with regard to (mathematical) ideals. Noted algebraist Irving Kaplansky called this work "revolutionary"; the publication gave rise to the term "
Noetherian ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
", and several other mathematical objects being called ''Noetherian (disambiguation), Noetherian''., p. 44–45.

# References

* * * * * * * * * * * * * *. Vol. II, Pure and Applied Mathematics 128, . * {{DEFAULTSORT:Ring Theory Ring theory, de:Ringtheorie ka:რგოლი (მათემატიკა) ro:Inel (algebră)