# Category: Ring theory

Automorphic number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base whose square "ends" in the same digits as the number itself.
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the for
Schreier domain
In abstract algebra, a Schreier domain, named after Otto Schreier, is an integrally closed domain where every nonzero element is primal; i.e., whenever x divides yz, x can be written as x = x1 x2 so t
Proofs of elementary ring properties
The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subt
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has
Topological ring
In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps: where carries the product topology. That means
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
Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: .
Sylvester domain
In mathematics, a Sylvester domain, named after James Joseph Sylvester by , is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R,
Matrix semialgebra
No description available.
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 i
In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form: * * when |x |
In mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -modules. It happens that substituting "left
Nilpotent
In mathematics, an element of a ring is called nilpotent if there exists some positive integer , called the index (or sometimes the degree), such that . The term was introduced by Benjamin Peirce in t
Fixed-point subring
In algebra, the fixed-point subring of an automorphism f of a ring R is the subring of the fixed points of f, that is, More generally, if G is a group acting on R, then the subring of R is called the
Near-ring
In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Dedekind-finite ring
A ring is said to be a Dedekind-finite ring if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided inverses in the ring are two-sided. These rings have also been cal
Kaplansky's conjectures
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kap
Signed-digit representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to a
Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indetermin
Opposite ring
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More ex
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 Wey
Rank ring
In mathematics, a rank ring is a ring with a real-valued rank function behaving like the rank of an endomorphism. John von Neumann introduced rank rings in his work on continuous geometry, and showed
Irreducible ideal
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
Pseudo-ring
In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: * A rng, i.e., a structure satisfying all the axioms of a ring except for the exis
Unipotent
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent matrix if
Semi-local ring
In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. The above definition is satisfied if R has a finite number of maximal right
Principal ideal ring
In 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
Clean ring
In mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than
Laurent polynomial
In mathematics, a Laurent polynomial (namedafter Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . La
Artinian ideal
In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings. Given a polynomial ring R = k[X1, ... Xn] where k is some field, an
Reduced ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that
Binomial ring
In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficients for x in the ring and n a positive integer. Binomial rings were introd
Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.Every finite field is an example of a finite ring, and the additive part of every finit
Regular ring
No description available.
Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of s
Witt vector
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt v
Projective line over a ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by
Classification of Clifford algebras
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadrati
Dense submodule
In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If N is a dense submodule of M, it may alternatively be said
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 ) if
Total ring of fractions
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may
Generalized Cohen–Macaulay ring
In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring of Krull dimension d > 0 that satisfies any of the following equivalent conditions: * For each integer , the lengt
Balanced module
In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian gro
Order (ring theory)
In mathematics, an order in the sense of ring theory is a subring of a ring , such that 1. * is a finite-dimensional algebra over the field of rational numbers 2. * spans over , and 3. * is a -latt
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.
Domain (ring theory)
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the o
V-ring (ring theory)
In mathematics, a V-ring is a ring R such that every simple R-module is injective. The following three conditions are equivalent: 1. * Every simple left (resp. right) R-module is injective 2. * The
Arithmetical ring
In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold: 1. * The localization of R at is a uniserial ring for every maximal ide
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
Integer-valued polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer n. Every polynomial with integer coefficients is integer
Necklace ring
In mathematics, the necklace ring is a ring introduced by Metropolis and Rota to elucidate the multiplicative properties of necklace polynomials.
Polynomial differential form
In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra: Varying n, it determines the simplicial commutative dg algebra: (each induces the ma
Quasiregular element
In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring. In this article, we primarily concern our
Uniform module
In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential su
Quasisymmetric function
In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring wit
Zorn ring
In mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent . named them after Max August Zorn, who studied a
Antiisomorphism
In category theory, a branch of mathematics, an antiisomorphism (or anti-isomorphism) between structured sets A and B is an isomorphism from A to the opposite of B (or equivalently from the opposite o
Subring
In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplica
Center (ring theory)
In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and is denoted as ; "Z" stands for the German word Ze
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
Zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rn
Matrix semiring
No description available.
Almost commutative ring
In algebra, a filtered ring A is said to be almost commutative if the associated graded ring is commutative. Basic examples of almost commutative rings involve differential operators. For example, the
Regular ideal
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal in a (possibly) non-unital ring A is said to be regular (or modular) if there
Composition ring
In mathematics, a composition ring, introduced in, is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation such that, for any three elem
Poisson ring
In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is
Weak dimension
In abstract algebra, the weak dimension of a nonzero right module M over a ring R is the largest number n such that the Tor group is nonzero for some left R-module N (or infinity if no largest such n
Idealizer
In abstract algebra, the idealizer of a subsemigroup T of a semigroup S is the largest subsemigroup of S in which T is an ideal. Such an idealizer is given by In ring theory, if A is an additive subgr
Fibonacci group
In mathematics, for a natural number , the nth Fibonacci group, denoted or sometimes , is defined by n generators and n relations: * * * * * * . These groups were introduced by John Conway in 19
Real closed ring
In mathematics, a real closed ring (RCR) is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the intege
Perfect field
In algebra, a field k is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over k has distinct roots. * Every irreducible polynomial over k is separable
Algebra (ring theory)
No description available.
Loewy ring
In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named
Matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a
Simple algebra (universal algebra)
In universal algebra, an abstract algebra A is called simple if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism with domain A is either injective or const
Square-free element
In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that is a unit of R.
Gelfand ring
In mathematics, a Gelfand ring is an associative ring R with identity such that if I and J are distinct right ideals then there are elements i and j such that iRj=0, i is not in I, and j is not in J.
Baer ring
In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, usin
Monoid ring
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Change of rings
In algebra, given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a left R-module M and a left S-module N, * , the induced module. * , the coinduce
Quasi-Frobenius ring
In mathematics, especially ring theory, the class of Frobenius rings and their generalizations are the extension of work done on Frobenius algebras. Perhaps the most important generalization is that o
Conductor (ring theory)
In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the correspon
Idempotent (ring theory)
In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication.
Product of rings
In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentw
Biquaternion
In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternio
Zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, This property is also known as the rule of zero product, the null factor law, the mult
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 th
Prime element
In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polyn
Goldman domain
In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman. An overring (i.e., an intermed
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: . Similarly, if M is a left R-module, then the associated graded module is the graded module
Overring
In mathematics, an overring B of an integral domain A is a subring of the field of fractions K of A that contains A: i.e., . For instance, an overring of the integers is a ring in which all elements a
Hochschild homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology
Leavitt path algebra
In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. The Leavitt path algebras generalize the and may also be considered as algebraic analogues of the graph
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 which allows a suitable generalization of the Eucli
Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either or . A modu
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras
Jacobson ring
In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by , based on a suggestion of
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that . The index set is usually the set of nonneg
Jacobson's conjecture
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special typ
Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. H
Prime ring
In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. This definition can be regarded as a simultaneous
Ring of polynomial functions
In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed
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. (N
Kleinian integer
In mathematical cryptography, a Kleinian integer is a complex number of the form , with m and n rational integers. They are named after Felix Klein. The Kleinian integers form a ring called the Kleini
Köthe conjecture
In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil i
Invariant basis number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the
Global dimension
In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or i
Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory a
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
Central polynomial
In algebra, a central polynomial for n-by-n matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at n-by-n matrices. That such p
Division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has
Free ideal ring
In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with
Representation ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classe
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x−1 belongs to D. Given a field F, if D is a subring of F
Semiprime ring
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical idea
Nakayama's conjecture
In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama. The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander a
Semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.
Polynomial identity ring
In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z⟨X1, X2, ..., XN⟩, over the ring of integers in N va
Non-integer base of numeration
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of is The numbers di are non-negative integers les
Artin algebra
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin. Every Artin algebra is an Artin ring.
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive
Ore condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative ring
Simplicial commutative ring
In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If A is a si
Derived algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded
Depth of noncommutative subrings
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain , whi
Glossary of ring theory
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Fo
Approximation property (ring theory)
In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in
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, an
Irreducible element
In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
Zero object (algebra)
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure,
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive id
Jaffard ring
In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for who first studied them
Bézout domain
In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a
Generalized Clifford algebra
In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these
Localization (commutative algebra)
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/
Noncommutative unique factorization domain
In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
Perfect ring
In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is
Seminormal ring
In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy , there is s with and . This definition was given by as a simplification of the original definition of . A b
Endomorphism ring
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endom
Free algebra
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"
Local ring
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defin
Coherent ring
In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extende
Triangular matrix ring
In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule.
Hilbert–Kunz function
In algebra, the Hilbert–Kunz function of a local ring (R, m) of prime characteristic p is the function where q is a power of p and m[q] is the ideal generated by the q-th powers of elements of the max
Centralizer and normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set of elements of G such that each member commutes with each element of S, or equiva
Ore algebra
In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. Th
Nakayama algebra
In algebra, a Nakayama algebra or generalized uniserial algebra is an algebra such that each left or right indecomposable projective module has a unique composition series. They were studied by Tadasi
Commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra
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 K-algebras, they generalize the real numbers, complex number
Semiprimitive ring
In algebra, a semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose Jacobson radical is zero. This is a type of ring more general than a semisimple ring, but where simple
Krull ring
In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimens
Simple ring
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and
Λ-ring
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natura
Annihilator (ring theory)
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of S. Over an integral domain,
SBI ring
In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI was introduced by Irving Kaplansky and stands f
Zero divisor
In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injectiv
Kasch ring
In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is
GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal
Buchsbaum ring
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence.A sequence of the maximal ideal is called a weak sequence if for all . They were intr
Semifield
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
Normal homomorphism
In algebra, a normal homomorphism is a ring homomorphism that is flat and is such that for every field extension L of the residue field of any prime ideal , is a normal ring.
Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is
Algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if A and B are algebras over a field (or commutative ring) K, it is a function such that for
Severi–Brauer variety
In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central
Divisibility (ring theory)
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the origina
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 desc
Novikov ring
In mathematics, given an additive subgroup , the Novikov ring of is the subring of consisting of formal sums such that and . The notion was introduced by Sergei Novikov in the papers that initiated th
Integral element
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial o
Irreducible ring
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways. * A (meet-)irreducible ring is one in which the intersection of two nonzero ideals i
Principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonze
Hopfian object
In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object
Ring extension
In commutative algebra, a ring extension is a ring homomorphism of commutative rings, which makes S an R-algebra. In this article, a ring extension of a ring R by an abelian group I is a pair of a rin
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existe
Stably finite ring
In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1.
Marot ring
In mathematics, a Marot ring, introduced by , is a commutative ring whose regular ideals are generated by regular elements.
Torsion-free module
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free i
Kurosh problem
In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases
Noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring
In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation: for every pair of elements and in the domain of For example, any linear map is additi
Minimal ideal
In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonz
Ore extension
In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood
Singular submodule
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (r
Hermite ring
In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to (p. 465), a ring is right Hermite if, for every two elements a and b of the ring, th
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R (associative, with 1, not necessarily commutative) such that for every element a in R there exists an x in R with a = axa. One may think of x as
Unit (ring theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative ide
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 f
Connected ring
In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions: * A possesses no non-trivial (that
Partially ordered ring
In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order on the underlying set A that is compatible with the ring operation
Boolean ring
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring g
Ring (mathematics)
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
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 theor
Integral closure of an ideal
In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that It is similar to the integral clo
Dedekind–Hasse norm
In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
Solenoid (mathematics)
In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms
Nilpotent algebra
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n
Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field
Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this or
Acceptable ring
In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein ring
Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categ
Universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal env
Unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem o
Hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is require
Complex-base system
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and in 1965).