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In mathematics, a noncommutative ring is a ring whose multiplication is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a commutative ring. Noncommutative algebra is the part of
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term ''noncommutative ring'' is used instead of ''ring'' to refer to a unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ''ring'' is used as a shortcut for ''commutative ring''. Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.


Examples

Some examples of noncommutative rings: * The matrix ring of ''n''-by-''n'' matrices over the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, where * Hamilton's
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s * Any
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
constructed from a group that is not
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
Some examples of rings that are not typically commutative (but may be commutative in simple cases): * The
free ring In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the po ...
\mathbb\langle x_1,\ldots,x_n\rangle generated by a finite set, an example of two non-equal elements being 2x_1x_2 + x_2x_1 \neq 3 x_1x_2 * The Weyl algebra A_n(\mathbb), being the ring of polynomial differential operators defined over affine space; for example, A_1(\mathbb) \cong \mathbb \langle x, y \rangle / (xy - yx - 1), where the ideal corresponds to the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
* The quotient ring \mathbb\langle x_1,\ldots,x_n\rangle / (x_ix_j - q_ x_jx_i), called a
quantum plane In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
, where q_ \in \mathbb * Any
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
can be described explicitly using an algebra presentation: given an \mathbb-vector space V of dimension with a quadratic form q: V\otimes V \to \mathbb, the associated Clifford algebra has the presentation \mathbb\langle e_1,\ldots, e_n \rangle /(e_ie_j + e_je_i - q(e_i,e_j)) for any basis e_1,\ldots,e_n of V, *
Superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. ...
s are another example of noncommutative rings; they can be presented as \mathbb _1,\ldots,x_nlangle \theta_1,\ldots, \theta_m \rangle / (\theta_i\theta_j + \theta_j\theta_i) * There are finite noncommutative rings: for example, the -by- matrices over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
, for . The smallest noncommutative ring is the ring of the upper triangular matrices over the field with two elements; it has eight elements and all noncommutative rings with eight elements are isomorphic to it or to its opposite.


History

Beginning with division rings arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes E. Artin,
Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
,
P. M. Cohn Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commu ...
, W. R. Hamilton, I. N. Herstein, N. Jacobson, K. Morita, E. Noether, Ø. Ore, J. Wedderburn and others.


Differences between commutative and noncommutative algebra

Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutative is the necessity to separately consider right ideals and left ideals. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left–right distinction does not exist.


Important classes


Division rings

A division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element ''a'' has a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
, i.e., an element ''x'' with . Stated differently, a ring is a division ring if and only if its
group of units In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for th ...
is the set of all nonzero elements. Division rings differ from fields only in that their multiplication is not required to be
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. However, by Wedderburn's little theorem all finite division rings are commutative and therefore
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".


Semisimple rings

A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of simple (irreducible) submodules. A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.


Semiprimitive rings

A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
is zero. This is a type of ring more general than a semisimple ring, but where
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
s still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of
primitive ring In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic ...
s, which are described by the
Jacobson density theorem In mathematics, more specifically non-commutative ring theory, 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 primitive ring can be ...
.


Simple rings

A simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of M(''n'',''R'') is of the form M(''n'',''I'') with ''I'' an ideal of ''R''), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns). According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s are rings of matrices over either the real numbers, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, or the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s. Any quotient of a ring by a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
is a simple ring. In particular, a field is a simple ring. A ring ''R'' is simple if and only if its opposite ring ''R''o is simple. An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.


Important theorems


Wedderburn's little theorem

Wedderburn's little theorem states that every
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite simple alternative ring is a field.


Artin–Wedderburn theorem

The Artin–Wedderburn theorem is a
classification theorem In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues relat ...
for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring ''R'' is isomorphic to a product of finitely many ''ni''-by-''ni'' matrix rings over division rings ''Di'', for some integers ''ni'', both of which are uniquely determined up to permutation of the index ''i''. In particular, any simple left or right
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
is isomorphic to an ''n''-by-''n'' matrix ring over a division ring ''D'', where both ''n'' and ''D'' are uniquely determined. As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result.
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
later generalized it to the case of Artinian rings.


Jacobson density theorem

The Jacobson density theorem is a theorem concerning
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
s over a ring . The theorem can be applied to show that any
primitive ring In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic ...
can be viewed as a "dense" subring of the ring of
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s of a vector space.Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by
Nathan Jacobson Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
s. More formally, the theorem can be stated as follows: :The Jacobson Density Theorem. Let be a simple right -module, , and a finite and -linearly independent set. If is a -linear transformation on then there exists such that for all in .


Nakayama's lemma

Let J(''R'') be the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
of ''R''. If ''U'' is a right module over a ring, ''R'', and ''I'' is a right ideal in ''R'', then define ''U''·''I'' to be the set of all (finite) sums of elements of the form ''u''·''i'', where · is simply the action of ''R'' on ''U''. Necessarily, ''U''·''I'' is a submodule of ''U''. If ''V'' is a
maximal submodule In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
of ''U'', then ''U''/''V'' is simple. So ''U''·J(''R'') is necessarily a subset of ''V'', by the definition of J(''R'') and the fact that ''U''/''V'' is simple. Thus, if ''U'' contains at least one (proper) maximal submodule, ''U''·J(''R'') is a proper submodule of ''U''. However, this need not hold for arbitrary modules ''U'' over ''R'', for ''U'' need not contain any maximal submodules. Naturally, if ''U'' is a Noetherian module, this holds. If ''R'' is Noetherian, and ''U'' is finitely generated, then ''U'' is a Noetherian module over ''R'', and the conclusion is satisfied. Somewhat remarkable is that the weaker assumption, namely that ''U'' is finitely generated as an ''R''-module (and no finiteness assumption on ''R''), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma. Precisely, one has the following. :Nakayama's lemma: Let ''U'' be a finitely generated right module over a ring ''R''. If ''U'' is a non-zero module, then ''U''·J(''R'') is a proper submodule of ''U''. A version of the lemma holds for right modules over non-commutative
unitary ring In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying proper ...
s ''R''. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem.


Noncommutative localization

Localization is a systematic method of adding multiplicative inverses to a ring, and is usually applied to commutative rings. Given a ring ''R'' and a subset ''S'', one wants to construct some ring ''R''* and
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition prese ...
from ''R'' to ''R''*, such that the image of ''S'' consists of '' units'' (invertible elements) in ''R''*. Further one wants ''R''* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
. The localization of ''R'' by ''S'' is usually denoted by ''S'' −1''R''; however other notations are used in some important special cases. If ''S'' is the set of the non zero elements of an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, then the localization is the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
and thus usually denoted Frac(''R''). Localizing non-commutative rings is more difficult; the localization does not exist for every set ''S'' of prospective units. One condition which ensures that the localization exists is the Ore condition. One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse ''D''−1 for a differentiation operator ''D''. This is done in many contexts in methods for
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s. There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches. The ''micro-'' tag is to do with connections with Fourier theory, in particular.


Morita equivalence

Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Two rings ''R'' and ''S'' (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over ''R'', ''R-Mod'', and the category of (left) modules over ''S'', ''S-Mod''. It can be shown that the left module categories ''R-Mod'' and ''S-Mod'' are equivalent if and only if the right module categories ''Mod-R'' and ''Mod-S'' are equivalent. Further it can be shown that any functor from ''R-Mod'' to ''S-Mod'' that yields an equivalence is automatically additive.


Brauer group

The Brauer group of a field ''K'' is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
whose elements are
Morita equivalence In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modul ...
classes of
central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simp ...
s of finite rank over ''K'' and addition is induced by the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of algebras. It arose out of attempts to classify
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a f ...
s over a field and is named after the algebraist
Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular represent ...
. The group may also be defined in terms of
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...
. More generally, the Brauer group of a scheme is defined in terms of
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local r ...
s.


Ore conditions

The Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, or more generally
localization of a ring In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractio ...
. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A domain that satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly.


Goldie's theorem

In mathematics, Goldie's theorem is a basic structural result in
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring ''R'' that has finite uniform dimension (also called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of ''R''. Goldie's theorem states that the
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
s, since by definition right Noetherian rings have the ascending chain condition on ''all'' right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of
prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only way ...
principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain.


See also

* Derived algebraic geometry *
Noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
* Noncommutative algebraic geometry * Noncommutative harmonic analysis *
Representation theory (group theory) In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...


Notes


References

* * *


Further reading

* * {{DEFAULTSORT:Non-Commutative Ring Ring theory