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In mathematics, element with a multiplicative inverse
a unit or invertible element 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
Unit_(ring_theory)
Submodule of a mathematical ring
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the
Ideal_(ring_theory)
Topics referred to by the same term
element Unit (ring theory), an element that is invertible with respect to ring multiplication Unit, a tuple of length 0; an empty tuple Statistical unit, a
Unit
Algebraic structure with addition and multiplication
on ring properties. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced
Ring_(mathematics)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field
Fundamental unit (number theory)
Fundamental_unit_(number_theory)
Vector space equipped with a bilinear product
is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra
Algebra_over_a_field
Generalization of vector spaces from fields to rings
this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital. An (R,S)-bimodule is an abelian group together
Module_(mathematics)
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This
Glossary_of_ring_theory
Concept in mathematical ring theory
article incorporates material from the Citizendium article "Divisibility (ring theory)", which is licensed under the Creative Commons Attribution-ShareAlike
Divisibility_(ring_theory)
Branch of number theory
supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's
Algebraic_number_theory
Topic in algebraic number theory
algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid
S-unit
Generalization of additive and multiplicative inverses
singular matrix, and cannot be inverted. Division ring Latin square property Loop (algebra) Unit (ring theory) Modular multiplicative inverse The usual definition
Inverse_element
Unique ring consisting of one element
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
Zero_ring
Branch of algebra that studies commutative rings
first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry
Commutative_algebra
(Mathematical) ring with a unique maximal ideal
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local
Local_ring
Algebraic ring without a multiplicative identity
(pronounced "rung" /rʌŋ/) or non-unital ring or pseudo-ring is an algebraic structure satisfying the same properties as a ring, but without assuming the existence
Rng_(algebra)
Ring without nonzero zero divisors
Zero-product property Divisor (ring theory) Integral domain Lam (2001), p. 3 Rowen (1994), p. 99. Some authors also consider the zero ring to be a domain: see Polcino
Domain_(ring_theory)
Algebraic construction
theory the elements of Z {\displaystyle \mathbb {Z} } are often called the "rational integers" because of this. The next simplest example is the ring
Ring_of_integers
In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map μ: E ∧ E → E and a unit map η: S → E, where S is the sphere
Ring_spectrum
Relationship between two functors abstracting many common constructions
elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative identity) into a ring. The most efficient
Adjoint_functors
Root of a quadratic polynomial with a unit leading coefficient
integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory. Medieval Indian mathematicians had
Quadratic_integer
Reduction of a ring by one of its ideals
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
Quotient_ring
Algebraic structure
number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group
Polynomial_ring
Branch of mathematics that studies algebraic structures
ring unit (ring theory), Idempotent, Nilpotent, Zero divisor Characteristic (algebra) Ring homomorphism, Algebra homomorphism Ring epimorphism Ring monomorphism
List of abstract algebra topics
List_of_abstract_algebra_topics
Algebraic structure
Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties
Noncommutative_ring
Ring built from other rings (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)
Product_of_rings
Commutative ring with a Euclidean division
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 Euclidean
Euclidean_domain
Structure in Ring Theory (Mathematics)
In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle
Jacobson_radical
Branch of mathematics
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology
K-theory
Algebraic ring that need not have additive negative elements
extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative
Semiring
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Numerous conjectures by mathematician Irving Kaplansky
from 3. Gardam, Giles (2021-02-23). "A counterexample to the unit conjecture for group rings". Annals of Mathematics. 194 (3): 967–979. arXiv:2102.11818
Kaplansky's_conjectures
Type of algebraic structure
product. The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring R such that M = ⨁ i ∈ N M i , {\displaystyle
Graded_ring
Algebraic structure
mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra
Commutative_ring
Commutative ring with no zero divisors other than zero
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every
Integral_domain
Set of finitely supported functions from a group to a ring
thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Let G {\displaystyle G} be a
Group_ring
Structure-preserving function between two rings
ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are said to be isomorphic. From the standpoint of ring theory,
Ring_homomorphism
In mathematics, element that equals its square
In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is
Idempotent_(ring_theory)
Category whose objects are rings and whose morphisms are ring homomorphisms
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
Category_of_rings
Direct sum of irreducible modules
as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a
Semisimple_module
Ideal ring structure
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good"[definition needed] elements of the ring. The first example of a
Radical_of_a_ring
Locally compact topological field
ring of integers O = { a ∈ F : | a | ≤ 1 } {\displaystyle {\mathcal {O}}=\{a\in F:|a|\leq 1\}} which is a discrete valuation ring, is the closed unit
Local_field
Token Ring network hub
media access unit (MAU), also known as a multistation access unit (MAU or MSAU), is a device to attach multiple network stations in a ring topology when
Media_access_unit
Mathematical ring with well-behaved ideals
right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics
Noetherian_ring
Type of integral domain
(a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as
Unique_factorization_domain
Mathematical concept
-module. Ring R ¯ {\textstyle {\bar {R}}} is Noetherian. An integrally closed local ring R {\textstyle R} is an integral domain or a ring whose non-unit elements
Overring
Ideal of a ring contained in no other ideal except the ring itself
In mathematics, more specifically in ring theory, a maximal ideal is a two-sided ideal that is maximal (with respect to set inclusion) amongst all proper
Maximal_ideal
In number theory, measure of non-unique factorization
the ring of integers of K {\displaystyle K} . The order of the group, which is finite, is called the class number of K {\displaystyle K} . The theory extends
Ideal_class_group
Supposition or system of ideas intended to explain something
theory — Perturbation theory — Potential theory — Probability theory — Ramsey theory — Rational choice theory — Representation theory — Ring theory —
Theory
noncommutative ring theory. Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 − r is a unit in R; that is
Quasiregular_element
Structure in some metamaterials
{\displaystyle \sigma } is the resistance of unit length of the sheets measured around the circumference. The split ring resonator and the metamaterial itself
Split-ring_resonator
Algebraic structure
operation on it. In mathematical analysis, the term also appears in the theory of one-parameter operator semigroups: see C0-semigroup. The binary operation
Semigroup
Construction within abstract algebra
In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero-divisors is the group of units of the ring, R × {\displaystyle
Total_ring_of_fractions
Free object in the category of associative algebras
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
Free_algebra
Mathematical structure in abstract algebra
x ∈ I ⇒ x* ∈ I and so on. *-rings are unrelated to star semirings in the theory of computation. A *-algebra A is a *-ring, with involution * that is an
*-algebra
Projective construction in ring theory
The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(A×), the center of A×, then the
Projective_line_over_a_ring
Algebraic structure with "nice" duality properties
especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind
Frobenius_algebra
Mathematical element
subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to
Integral_element
Ring that is also a vector space or a module
commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative
Associative_algebra
Sum in algebraic number theory
commutative ring R, ψ is a group homomorphism of the additive group R+ into the unit circle, and χ is a group homomorphism of the unit group R× into the unit circle
Gauss_sum
Concept in number theory
In number theory, the adele ring is a construction that combines all local versions of a global field into one object. For the rational numbers, these
Adele_ring
Möbius transformation generalized to rings other than the complex numbers
must be a unit of the domain (that is 1 or −1 in the case of integers). In the most general setting, the a, b, c, d and z are elements of a ring, such as
Linear fractional transformation
Linear_fractional_transformation
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if all of its elements are nilpotent
Nil_ideal
Number-theoretic concept
_{p}} is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In
Profinite_integer
Endomorphism algebra of an abelian group
these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0 : x ↦ 0 {\textstyle 0:x\mapsto 0} as additive identity
Endomorphism_ring
Construction of a ring of fractions
provides a natural link to sheaf theory. In fact, the term localization originated in algebraic geometry: if R is a ring of functions defined on some geometric
Localization (commutative algebra)
Localization_(commutative_algebra)
Algebraic structure with an associative operation and an identity element
between category theory and monoids see below. The set of homeomorphism classes of compact surfaces with the connected sum. Its unit element is the class
Monoid
field may be a topological ring which is not a field. The group of units R × {\displaystyle R^{\times }} of a topological ring R {\displaystyle R} is a
Topological_ring
Algebraic structure
ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is
Principal_ideal_domain
Element in a ring whose some power is 0
generally, the sum of a unit element and a nilpotent element is a unit when they commute. The nilpotent elements from a commutative ring R {\displaystyle R}
Nilpotent
Generalization of algebraic variety
"varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck
Scheme_(mathematics)
particularly ring theory and modulus theory. The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory. Birch–Tate
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element. The theorem
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
In algebra, expression of an ideal as the intersection of ideals of a specific type
mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection
Primary_decomposition
Concept in ring theory
where R {\displaystyle R} is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck
Azumaya_algebra
Type of module over a ring
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
Simple_module
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal i {\displaystyle {\mathfrak
Regular_ideal
Fraction with denominator a power of two
dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted Z [ 1 2 ] {\displaystyle
Dyadic_rational
Operation in algebra and mathematics
In category theory, a branch of mathematics, a monad is a triple ( T , η , μ ) {\displaystyle (T,\eta ,\mu )} consisting of a functor T from a category
Monad_(category_theory)
Algebraic structure
denominator has nonzero constant term. Again this composition ring has no multiplicative unit; if R is a field, it is in fact a subring of the formal power
Composition_ring
Number in {..., –2, –1, 0, 1, 2, ...}
integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, integers are sometimes called rational integers
Integer
Cohomology class
mathematics, a highly structured ring spectrum or A ∞ {\displaystyle A_{\infty }} -ring is an object in homotopy theory encoding a refinement of a multiplicative
Highly structured ring spectrum
Highly_structured_ring_spectrum
In algebra, element without non-trivial factors
commutative ring. Let R {\displaystyle R} be an integral domain. An element a ∈ R {\displaystyle a\in R} is irreducible if it is not a unit and whenever
Irreducible_element
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
Primitive_ring
Result pertaining to division rings
and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings K ⊆ D such that xKx−1 is contained in K for every x not
Cartan–Brauer–Hua_theorem
Concept in modular arithmetic
of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n. The order of a modulo n is sometimes written
Multiplicative_order
Mathematical construct
of Morse theory using a closed one-form instead of a function. The notion is used in quantum cohomology, among the others. The Novikov ring Nov ( Γ
Novikov_ring
Commutative ring with a well behaved theory of prime factorization
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
Krull_ring
Algebraic structure in mathematics
(same as the ring product) and use x ∨ y for the join, given in terms of ring notation (given just above) by x + y + xy. In set theory and logic it is
Boolean_ring
Annihilator of a simple module
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined
Primitive_ideal
Algebraic ring classification
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. (Lam 2001, p. §20)(Mikhalev
Semi-local_ring
Mathematical topics based on the works of George Boole
the network Boolean processor, a 1-bit variable computing unit Boolean ring, a mathematical ring for which x2 = x for every element x Boolean satisfiability
Boolean
Ideal in a ring which has properties similar to prime elements
algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers
Prime_ideal
below. A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen, P.M. Cohn
Serial_module
Concept in algebra
that any ring satisfying the first three conditions satisfies the fourth: take Γ to be the quotient K×/D× of the unit group of K by the unit group of
Valuation_ring
Net in a normed algebra
particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts
Approximate_identity
Algebraic structure generalizing Boolean rings
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
Clean_ring
Mathematical term in group theory
In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p ∞ {\displaystyle p^{\infty }} -group, Z ( p ∞ ) {\displaystyle
Prüfer_group
Mathematical ring whose elements are matrices
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
Matrix_ring
Theorem characterizing the automorphisms of simple rings
In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the
Skolem–Noether_theorem
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