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(Mathematical) ring with a unique maximal ideal
specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the
Local_ring
Type of ring in commutative algebra
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
Regular_local_ring
In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain
Unibranch_local_ring
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
Algebraic structure with addition and multiplication
ring Lie ring Local ring Noetherian and artinian rings Ordered ring Poisson ring Reduced ring Regular ring Ring of periods SBI ring Valuation ring and discrete
Ring_(mathematics)
In algebraic geometry, a Noetherian local ring R is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec(R) − m) of its spectrum
Parafactorial_local_ring
Type of commutative ring in mathematics
mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under
Cohen–Macaulay_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
deviations of a local ring R are certain invariants εi(R) that measure how far the ring is from being regular. The deviations εn of a local ring R with residue
Deviation_of_a_local_ring
Local ring in commutative algebra
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many
Gorenstein_ring
catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings A local complete intersection ring is a Noetherian
Complete_intersection_ring
Concept in abstract algebra
conditions: R {\displaystyle R} is a local ring, a principal ideal domain, and not a field. R {\displaystyle R} is a valuation ring with a value group isomorphic
Discrete_valuation_ring
Ring in abstract algebra
mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided)
Artinian_ring
Branch of algebra
integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division
Ring_theory
geometrically regular local ring. acceptable ring Acceptable rings are generalizations of excellent rings, with the conditions about regular rings in the definition
Glossary of commutative algebra
Glossary_of_commutative_algebra
Local ring in which Hensel's lemma holds
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by Azumaya (1951), who named them
Henselian_ring
Concept in algebra
particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially
Valuation_ring
Real numbers adjoined with a nil-squaring element
dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. Dual numbers
Dual_number
In mathematics, dimension of a ring
A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal
Krull_dimension
states that a projective module over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Construction of a ring of fractions
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 fractions
Localization (commutative algebra)
Localization_(commutative_algebra)
an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent). The following rings are analytically unramified:
Analytically_unramified_ring
equicharacteristic Noetherian local ring is a ring of formal power series over a field. (Equicharacteristic means that the local ring and its residue field have
Cohen_structure_theorem
Sheaf of rings in mathematics
mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that
Ringed_space
Topics referred to by the same term
small neighborhoods of points Local ring, type of ring in commutative algebra Pub, a drinking establishment, known as a "local" to its regulars All pages
Local
In algebra, completion w.r.t. powers of an ideal
of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring. There
Completion_of_a_ring
Type of Grothendieck topology on the category of schemes
correct analog of the local ring at x is formed by taking the limit over a strictly larger family. The correct analog of the local ring at x for the étale
Étale_topology
ideal are N-2 rings). A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring, but this concept
Nagata_ring
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)
Concept in number theory
adele ring is a construction in number theory that combines all local versions of a global field into one object. For the rational numbers, these local versions
Adele_ring
Representation of an algebra as a free module
finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in
Hironaka_decomposition
terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points
Geometrically_regular_ring
Set of a ring's prime ideals
Zariski topology. The spectrum of a ring is also equipped with a structure of ringed space, that is, commutative rings are associated to every point and
Spectrum_of_a_ring
Direct summand of a free module (mathematics)
other rings over which they are true. For example, the implication labeled "local ring or PID" is also true for (multivariate) polynomial rings over a
Projective_module
Direct sum of irreducible modules
its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups
Semisimple_module
Theorem in algebra mathematics
varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese
Nakayama's_lemma
Mathematical ring with well-behaved ideals
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
Noetherian_ring
Saturn has the most extensive and complex ring system of any planet in the Solar System. The rings consist of particles in orbit around the planet, ranging
Rings_of_Saturn
Manifold upon which it is possible to perform calculus
sheaf of rings on Rn. The stalk Op for p ∈ Rn consists of germs of functions near p, and is an algebra over R. In particular, this is a local ring whose
Differentiable_manifold
Algebraic structure in ring theory
principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences
Flat_module
Type of integral domain
Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD. The converse of this is not true: there are Noetherian local rings that
Unique_factorization_domain
French mathematician (1928–2014)
Excellent ring Fibred category – Concept in category theory Formally smooth map Fundamental groupoid Fundamental group scheme Gorenstein ring – Local ring in
Alexander_Grothendieck
inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian
Catenary_ring
Topics referred to by the same term
Look up ring in Wiktionary, the free dictionary. (The) Ring(s) may refer to: Ring (jewellery), a round band, usually made of metal, worn as ornamental
Ring
Algebraic structure
explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well. A ring whose localizations at all
Integrally_closed_domain
Branch of algebra that studies commutative rings
"localization of a ring", "local ring", "regular ring". An affine algebraic variety corresponds to a prime ideal in a polynomial ring, and the points of
Commutative_algebra
Home security products manufacturer
allows users to discuss local safety and security issues, and share footage captured with Ring products. Via Neighbors, Ring could also provide footage
Ring_(company)
Study of dimension in algebraic geometry
of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological
Dimension_theory_(algebra)
Invariant of rings and modules
case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension
Depth_(ring_theory)
In mathematics, element with a multiplicative inverse
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 exists
Unit_(ring_theory)
In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient
Analytically_normal_ring
Local theory of several complex variables
as the Weierstrass preparation theorem, for the ring of formal power series over complete local rings A: for any power series f = ∑ n = 0 ∞ a n t n ∈
Weierstrass preparation theorem
Weierstrass_preparation_theorem
Well-behaved sequence in a commutative ring
is a zero-divisor in the ring C[x,y,z]/(y(1-x)) since z(1-x), y ≠ 0 but z(1-x)y = 0. However, if R is a Noetherian local ring and the elements ri are in
Regular_sequence
Algebra term
ring (Z/2Z)[F*/F*2] if q ≡ 1 mod 4. The Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to the group ring
Witt_group
Commutative ring with no zero divisors other than zero
regular local ring is an integral domain. In fact, a regular local ring is a UFD. The following rings are not integral domains. The zero ring (the ring in
Integral_domain
Endomorphism algebra of an abelian group
endomorphism ring being a local ring. For a semisimple module, the endomorphism ring is a von Neumann regular ring. The endomorphism ring of a nonzero
Endomorphism_ring
Technology for computer networking
Token Ring is a physical and data link layer computer networking technology used to build local area networks. It was introduced by IBM in 1984, and standardized
Token_Ring
Concept in commutative algebra
Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined
Excellent_ring
In mathematics, a module that has a basis
the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R
Free_module
Equivalence class of objects sharing local properties at a point in a topological space
The property that rings of germs are local rings is axiomatized by the theory of locally ringed spaces. The types of local rings that arise, however
Germ_(mathematics)
Theorem of algebraic geometry and commutative algebra
including: A normal local ring is unibranch, which is a variation of the statement that the transform of a normal point is connected. The local ring of a normal
Zariski's_main_theorem
Computer network that connects devices over a limited area
technologies used for local area networks; historical network technologies include ARCNET, Token Ring, and LocalTalk. A local area network allows multiple
Local_area_network
Axiom specifying the requisites of a sheaf on a topological space
into a category of local rings. It is the stalks of the sheaf that are local rings, not the collections of sections (which are rings, but in general are
Gluing_axiom
Branch of mathematics that studies algebraic structures
regular ring Quasi-Frobenius ring Hereditary ring, Semihereditary ring Local ring, Semi-local ring Discrete valuation ring Regular local ring Cohen–Macaulay
List of abstract algebra topics
List_of_abstract_algebra_topics
Concept in algebraic geometry
the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X)
Normal_scheme
Locally compact topological field
non-Archimedean local field F {\displaystyle F} with absolute value | ⋅ | {\displaystyle |\cdot |} , the following objects are important: its ring of integers
Local_field
Structure in algebraic geometry
Nisnevich topology, the local rings are Henselian, and a finite cover of a Henselian ring is given by a product of Henselian rings, showing exactness. If
Nisnevich_topology
Field arising from a quotient ring by a maximal ideal
R} is a commutative local ring, with maximal ideal m {\displaystyle {\mathfrak {m}}} . Then the residue field is the quotient ring R / m {\displaystyle
Residue_field
Traditional Irish ring
A Claddagh ring (Irish: fáinne an Chladaigh) is a traditional Irish ring with three primary features: a heart to represent love, a crown to represent
Claddagh_ring
Point where a mathematical object behaves irregularly
varieties and schemes: A point is singular if the local ring at this point is not a regular local ring. Catastrophe theory Defined and undefined Degeneracy
Singularity_(mathematics)
Mathematical element
In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over
Integral_element
In algebra, integer associated to a module
In algebra, the length of a module over a ring R {\displaystyle R} is a generalization of the dimension of a vector space which measures its size. page
Length_of_a_module
Generalizations of codimension-1 subvarieties of algebraic varieties
varieties, and results on local rings rather than projective varieties. In particular, if R is a complete intersection local ring which is factorial in codimension
Divisor_(algebraic_geometry)
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
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
Concept in commutative algebra
Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that ⋂ n > 0 a n = 0 {\displaystyle \bigcap
I-adic_topology
Theorem in algebra
and Noetherian modules over a complete Noetherian local ring. In the special case when the local ring contains a field mapping to the residue field it
Matlis_duality
Mathematical object in abstract algebra
RP-injective hull of R/P. In other words, it suffices to consider local rings. The endomorphism ring of the injective hull of R/P is the completion R ^ P {\displaystyle
Injective_module
semiperfect rings include: Left (right) perfect rings. Local rings. Left (right) Artinian rings. Finite dimensional k-algebras. Since a ring R is semiperfect
Perfect_ring
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
Jacobson_ring
"semi-local ring" which now means something different, and named "Zariski rings" by Pierre Samuel (1953). Examples of Zariski rings are noetherian local rings
Zariski_ring
Construction in homological algebra
tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts
Koszul_complex
concepts of homological algebra. Let R be a Noetherian, commutative, regular local ring and let P and Q be prime ideals of R. Serre defined the intersection multiplicity
Serre's multiplicity conjectures
Serre's_multiplicity_conjectures
Minimal element in the set of prime ideals ordered by inclusion
ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use
Minimal_prime_ideal
Tangent spaces in algebraic geometry
{\displaystyle R=C_{0}^{1}(\mathbf {R} )} to be the ring of germs of such functions at the origin. Then R is a local ring, and its maximal ideal m consists of all
Zariski_tangent_space
Discrete valuation field
multi-dimensional local fields. On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is
Higher_local_field
Infinite sum that is considered independently from any notion of convergence
{\displaystyle R[[X]]} as a product topology. The ring of formal power series with coefficients in a complete local ring satisfies the Weierstrass preparation theorem
Formal_power_series
Concept in algebraic geometry
The local cohomology module H ( x ) 1 ( K [ x ] ) {\displaystyle H_{(x)}^{1}(K[x])} (where K [ x ] {\displaystyle K[x]} is the coordinate ring of A K
Local_cohomology
Concept in algebraic geometry
presentation and is formally étale for maps from local rings, that is: Let A {\displaystyle A} be a local ring and J {\displaystyle J} be an ideal of A {\displaystyle
Étale_morphism
In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only
Analytically_irreducible_ring
Folk song
"Ring a Ring o' Roses", also known as "Ring a Ring o' Rosie" or "Ring Around the Rosie", is a nursery rhyme, folk song, and playground game. Descriptions
Ring_a_Ring_o'_Roses
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined
G-ring
property is said to be local if it can be detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being
Local_property
regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points is an open subset, and a J-2 ring is a ring such
J-2_ring
Local government area in New South Wales, Australia
Ku-ring-gai Council is a local government area in Northern Sydney (Upper North Shore), in the state of New South Wales, Australia. The area is named after
Ku-ring-gai_Council
Rings admitting weak inverses
regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element a of a ring is called
Von_Neumann_regular_ring
Number of times an object must be counted for making true a general formula
the local ring at this component of the coordinate ring of the intersection has only one prime ideal, and is therefore an Artinian ring. This ring is thus
Multiplicity_(mathematics)
Ring of stars in the Milky Way galaxy
The Gould Belt is a local ring of stars in the Milky Way, tilted away from the galactic plane by about 16–20 degrees, first reported by John Herschel and
Gould_Belt
Tool in mathematical dimension theory
ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. The filtration of a local
Hilbert series and Hilbert polynomial
Hilbert_series_and_Hilbert_polynomial
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent
Grothendieck_local_duality
LOCAL RING
LOCAL RING
Boy/Male
English American French
Faithful; unswerving.
Boy/Male
Arabic
Loyal
Boy/Male
British, English
Loyal
Boy/Male
Italian Greek
Loyal.
Boy/Male
English American
Loyal.
Boy/Male
British, English
Loyal
Boy/Male
American, Australian, British, English, French
Faithful; True
Boy/Male
American, British, English, Italian
Loyal
Boy/Male
Irish Welsh
Loyal.
Girl/Female
Indian
Loyal
Boy/Male
Indian
Loyal
Boy/Male
Irish Welsh
Loyal.
Boy/Male
American, British, English
Loyal
Girl/Female
Arabic, Muslim
Loyal
Boy/Male
British, English
Loyal
Boy/Male
Irish American Welsh
Loyal.
Boy/Male
Irish
Loyal.
Girl/Female
French
Loyal.
Girl/Female
Muslim
Loyal
Boy/Male
Irish
Loyal.
LOCAL RING
LOCAL RING
Male
Greek
(Î Ïίαμος) Greek name, possibly derived from the Luwian compound priimuua, PRIAMOS means "exceptionally courageous." Luwian is closely related to Hittite. In mythology, this is the name of a king of Troy.
Girl/Female
Arabic, Indian
Woman
Boy/Male
Gaelic American Irish
Little king.
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lotus
Girl/Female
American, Australian, Chinese, Greek, Hawaiian, Hebrew, Portuguese
She who Hears; God has Heard; Listening; Hearkening
Boy/Male
Indian
The Sun
Girl/Female
English American
Place name; a London district.
Surname or Lastname
English
English : from a pet form of Peter.
Boy/Male
Muslim
Husband. Crown.
Surname or Lastname
English
English : habitational name from Beeley in Derbyshire, which is named with the Old English personal names Bēage (female) or Bēga (male) + lēah ‘woodland clearing’.
LOCAL RING
LOCAL RING
LOCAL RING
LOCAL RING
LOCAL RING
a.
Belonging to,or concerning, a focus; as, a focal point.
a.
Alt. of Loral
n.
A vocal sound; specifically, a purely vocal element of speech, unmodified except by resonance; a vowel or a diphthong; a tonic element; a tonic; -- distinguished from a subvocal, and a nonvocal.
a.
Faithful; loyal.
v. t.
To divide according to gepgraphical sections or local interests.
n.
A district or local division, as of a province.
n.
A man who has a right to vote in certain elections.
n.
Vocal expression; articulation; speech.
a.
Uttered or modulated by the voice; oral; as, vocal melody; vocal prayer.
a.
Loyal.
n.
A local European measure of length. See Canna.
a.
Consisting of, or characterized by, voice, or tone produced in the larynx, which may be modified, either by resonance, as in the case of the vowels, or by obstructive action, as in certain consonants, such as v, l, etc., or by both, as in the nasals m, n, ng; sonant; intonated; voiced. See Voice, and Vowel, also Guide to Pronunciation, // 199-202.
a.
Confined to no zone or region; not local.
a.
Of or pertaining to a particular place, or to a definite region or portion of space; restricted to one place or region; as, a local custom.
n.
A local name of the burbot.
n.
A train which receives and deposits passengers or freight along the line of the road; a train for the accommodation of a certain district.
a.
Faithful; loyal; true.
a.
Of or pertaining to a vowel; having the character of a vowel; vowel.
n.
On newspaper cant, an item of news relating to the place where the paper is published.
n.
A principle, practice, form of speech, or other thing of local use, or limited to a locality.