In

_{0}(R). Note that K_{0}(R) = K_{0}(S) if two commutative rings ''R'', ''S'' are Morita equivalent.

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 branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

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
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 ana ...

s are much better understood than noncommutative ones. Algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

and algebraic number theory
Algebraic number theory is a branch of number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...

, 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 ''Satz
' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...

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 'cr ...

, 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 Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian ring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s.
For the definitions of a ring and basic concepts and their properties, see ''Ring (mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

''. The definitions of terms used throughout ring theory may be found in ''Glossary of ring theory
Ring theory is the branch of mathematics in which ring (mathematics), rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.
For the items in commutative ...

''.
Commutative rings

A ring is called ''commutative'' if its multiplication iscommutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. 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 branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s. Commutative rings are also important in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. In commutative ring theory, numbers are often replaced by ideals
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...

, and the definition of the prime ideal
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. ...

tries to capture the essence of prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. Integral domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...

s are integral domains in which the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

can be carried out. Important examples of commutative rings can be constructed as rings of polynomial
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). I ...

s and their factor rings. Summary: Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...

⊂ principal ideal domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

⊂ unique factorization domain
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 ...

⊂ integral domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

⊂ commutative ring
In ring theory
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 ana ...

.
Algebraic geometry

Algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

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 ''Satz
' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

that establishes a one-to-one correspondence between the points of an algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematic ...

, and the maximal ideal
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s of its coordinate ring
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

. 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
Alexander Grothendieck (; ; ; 28 March 1928 – 13 November 2014) was a mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbe ...

completed this by introducing schemes, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely,
the spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum
Continuum may refer to:
* Continuum (measurement)
Continuum theories or models expla ...

of a commutative ring is the space of its prime ideals equipped with Zariski topology
In algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained () ...

, and augmented with a sheaf
Sheaf may refer to:
* Sheaf (agriculture)
A sheaf (/ʃiːf/) is a bunch of cereal-crop stems bound together after reaping, traditionally by sickle, later by scythe or, after its introduction in 1872, by a mechanical reaper-binder.
Traditional ...

of rings. These objects are the "affine schemes" (generalization of affine varieties
Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to:
*Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology
*Affine cipher, a special case of the more general substitution cipher
*Aff ...

), 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

by gluing together the charts
A chart is a graphical representation
Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images such as drawings and photographs, and c ...

of an atlas
Blaeu's world map, originally prepared by Joan Blaeu for his ''Atlas Maior">Joan_Blaeu.html" ;"title="world map, originally prepared by Joan Blaeu">world map, originally prepared by Joan Blaeu for his ''Atlas Maior'', published in the first b ...

.
Noncommutative rings

Noncommutative rings resemble rings ofmatrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object.
Fo ...

in many respects. Following the model of algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, 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 algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s (rings that are also vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s) are often studied via their categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

of modules. A module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modula ...

over a ring is an abelian group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that the ring acts on as a ring of endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...

s, very much akin to the way field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

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 matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object.
Fo ...

or more generally by rings of endomorphisms of abelian groups or modules, and by monoid ring
In abstract algebra, a monoid ring is a ring (algebra), ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group (mathematics), group.
Definition
Let ''R'' be a ring and let ''G'' be a monoid. The mono ...

s.
Representation theory

Representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

that draws heavily on non-commutative rings. It studies 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 elements as linear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, and studies
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 ...

over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object.
Fo ...

and the algebraic operation
In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...

s in terms of matrix addition
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 ...

and matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, 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 groups
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

, associative algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s and Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. The most prominent of these (and historically the first) is the 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 theorems for rings *Nakayama's lemma In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

Structure theorems
*The Artin–Wedderburn theoremIn 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 i ...

determines the structure of semisimple ring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s
*The Jacobson density theoremIn mathematics, more specifically non-commutative ring theory, Abstract algebra, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring .
The theorem can be applied to show that any primiti ...

determines the structure of primitive rings
* Goldie's theorem determines the structure of semiprime
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

Goldie rings
*The Zariski–Samuel theorem In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

determines the structure of a commutative principal ideal ringIn mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called princ ...

*The Hopkins–Levitzki theorem gives necessary and sufficient conditions for a Noetherian ring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

to be an Artinian ringIn abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first dis ...

* 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 theoremIn 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 ha ...

states that finite domains are fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

Other
*The Skolem–Noether theorem characterizes the automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s of simple ring In abstract algebra, a branch of mathematics, a simple ring is a zero ring, non-zero ring (mathematics), ring that has no two-sided ideal (ring theory), ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if a ...

s
Structures and invariants of rings

Dimension of a commutative ring

In this section, ''R'' denotes a commutative ring. TheKrull dimension
In 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 ...

of ''R'' is the supremum of the lengths ''n'' of all the chains of prime ideals $\backslash mathfrak\_0\; \backslash subsetneq\; \backslash mathfrak\_1\; \backslash subsetneq\; \backslash cdots\; \backslash subsetneq\; \backslash mathfrak\_n$. It turns out that the polynomial ring $k;\; href="/html/ALL/s/\_1,\_\backslash cdots,\_t\_n.html"\; ;"title="\_1,\; \backslash cdots,\; t\_n">\_1,\; \backslash cdots,\; t\_n$Hilbert polynomialIn commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...

).
A commutative ring ''R'' is said to be catenary
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

if for every pair of prime ideals $\backslash mathfrak\; \backslash subset\; \backslash mathfrak\text{'}$, there exists a finite chain of prime ideals $\backslash mathfrak\; =\; \backslash mathfrak\_0\; \backslash subsetneq\; \backslash cdots\; \backslash subsetneq\; \backslash mathfrak\_n\; =\; \backslash mathfrak\text{'}$ 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 $\backslash mathfrak$ and $\backslash mathfrak\text{'}$ 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 $\backslash mathfrak$,
:$\backslash operatornameR\; =\; \backslash operatorname\backslash mathfrak\; +\; \backslash operatornameR/\backslash mathfrak$
where $\backslash operatorname\backslash mathfrak$ is the height
Height is measure of vertical distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can ...

of $\backslash mathfrak$.
If ''R'' is an integral domain that is a finitely generated ''k''-algebra, then its dimension is the transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' /''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

of its field of fractions over ''k''. If ''S'' is an integral extension In 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 n ...

of a commutative ring ''R'', then ''S'' and ''R'' have the same dimension.
Closely related concepts are those of depth
Depth(s) may refer to:
Science
* Three-dimensional space
* Depth (ring theory), an important invariant of rings and modules in commutative and homological algebra
* Depth in a well, the measurement between two points in an oil well
* Color depth ...

and global dimensionIn 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 ...

. 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
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebraic geometry, algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly w ...

. A regular local ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ide ...

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
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
*Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chromos ...

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 thecategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

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 $\backslash mathbf(R)$ the set of isomorphism classes of finitely generatedprojective module
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s over ''R''; let also $\backslash mathbf\_n(R)$ subsets consisting of those with constant rank ''n''. (The rank of a module ''M'' is the continuous function $\backslash operatornameR\; \backslash to\; \backslash mathbb,\; \backslash ,\; \backslash mathfrak\; \backslash mapsto\; \backslash dim\; M\; \backslash otimes\_R\; k(\backslash mathfrak)$.) $\backslash mathbf\_1(R)$ is usually denoted by Pic(''R''). It is an abelian group called the Picard 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 h ...

of ''R''. If ''R'' is an integral domain with the field of fractions ''F'' of ''R'', then there is an exact sequence of groups:
:$1\; \backslash to\; R^*\; \backslash to\; F^*\; \backslash overset\backslash to\; \backslash operatorname(R)\; \backslash to\; \backslash operatorname(R)\; \backslash to\; 1$
where $\backslash operatorname(R)$ is the set of fractional ideal
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 of ''R''. If ''R'' is a regular
The term regular can mean normal or in accordance with rules. It may refer to:
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* Regular (Badfinger song), "Regular" (Badfinger song)
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domain (i.e., regular at any prime ideal), then Pic(R) is precisely the divisor class group
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier ...

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 integersIn 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 ha ...

, 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 $\backslash mathbf(R)$; this results in a commutative ring KStructure of noncommutative rings

The structure of anoncommutative 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). ...

is more complicated than that of a commutative ring. For example, there exist simple
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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 ringIn 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 i ...

, 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 radicalIn 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 ha ...

of a ring; that is, the intersection of all right/left annihilators of simple
Simple or SIMPLE may refer to:
* Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...

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'' matrices over a field is noncommutative despite its natural occurrence in geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and many parts of mathematics. More generally, endomorphism ringIn abstract algebra, the endomorphisms of an abelian group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...

s of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...

.
One of the best known noncommutative rings is the division ring of quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion a ...

.
Applications

The ring of integers of a number field

The coordinate ring of an algebraic variety

If ''X'' is anaffine algebraic variety
Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to:
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* Affine cipher, a special case of the more general substitution cipher
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, then the set of all regular functions on ''X'' forms a ring called the coordinate ring
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

of ''X''. For a projective variety
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...

, there is an analogous ring called the homogeneous coordinate ring In algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), a ...

. 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 ''Satz
' (German for ''sentence'', ''movement'', ''set'', ''setting'') is any single member of a musical piece, which in and of itself displays ...

or scheme-theoretic constructions (i.e., Spec and Proj).
Ring of invariants

A basic (and perhaps the most fundamental) question in the classicalinvariant theory
Invariant theory is a branch of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...

is to find and study polynomials in the polynomial ring $k;\; href="/html/ALL/s/.html"\; ;"title="">$ring of symmetric polynomials
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring (mathematics), ring serves as universal ...

: symmetric polynomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s are polynomials that are invariant under permutation of variable. The fundamental theorem of symmetric polynomials
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symm ...

states that this ring is $R;\; href="/html/ALL/s/sigma\_1,\_\backslash ldots,\_\backslash sigma\_n.html"\; ;"title="sigma\_1,\; \backslash ldots,\; \backslash sigma\_n">sigma\_1,\; \backslash ldots,\; \backslash sigma\_n$History

Commutative ring theory originated in algebraic number theory, algebraic geometry, andinvariant theory
Invariant theory is a branch of abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...

. 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
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 ...

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
Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin
, name_Latin = Collegium Sanctae et Individuae Trinitatis Reg ...

put forth the quaternion
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s and biquaternionIn abstract algebra, the biquaternions are the numbers , where , and are complex number
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called ...

s; James Cockle
Sir James Cockle Royal Society, FRS Royal Astronomical Society, FRAS Cambridge Philosophical Society, FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son o ...

presented tessarine
In abstract algebra, a bicomplex number is a pair of complex number
In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, ...

s and coquaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) ...

s; and William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

was an enthusiast of split-biquaternionIn 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 ha ...

s, which he called ''algebraic motors''. These noncommutative algebras, and the non-associative Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s, were studied within universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spa ...

before the subject was divided into particular mathematical 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 ...

types. One sign of re-organization was the use of direct sums to describe algebraic structure.
The various hypercomplex numbers were identified with matrix ring
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s by Joseph Wedderburn
Joseph Henry Maclagan Wedderburn Fellow of the Royal Society of Edinburgh, FRSE Fellow of the Royal Society of London, FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his caree ...

(1908) and Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

(1928). Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian ringIn abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first dis ...

s.
In 1920, Emmy Noether
Amalie Emmy Noether Emmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called ''Idealtheorie in Ringbereichen'', analyzing ascending chain conditionIn mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings.Jacobson (2009), p. 1 ...

s with regard to (mathematical) ideals. Noted algebraist Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. htt ...

called this work "revolutionary"; the publication gave rise to the term "Noetherian ring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

", and several other mathematical objects being called ''NoetherianIn mathematics, the adjective
In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...

''., p. 44–45.
Notes

References

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