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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
older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple
Geometrically_regular_ring
(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 commutative ring in mathematics
assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central
Cohen–Macaulay_ring
Rings admitting weak inverses
Neumann 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
Algebraic structure
ring over k. Broadly speaking, regular local rings are somewhat similar to polynomial rings. Regular local rings are UFD's. Discrete valuation rings are
Commutative_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
Local ring in commutative algebra
intersection rings ⊃ regular local rings A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring
Gorenstein_ring
Well-behaved sequence in a commutative ring
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This
Regular_sequence
Absolutely regular is an alternative term for geometrically regular. 6. An absolutely simple point is one with a geometrically regular local ring. acceptable
Glossary of commutative algebra
Glossary_of_commutative_algebra
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)
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
Cohen–Macaulay ring. A regular local ring is an example of a Cohen–Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has
Ring_theory
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
inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian
Catenary_ring
Commutative ring with no zero divisors other than zero
A 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
Integral_domain
In mathematics, dimension of a ring
is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local ring is an example of such a ring. A Noetherian integral domain
Krull_dimension
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
Serre's multiplicity conjectures
Serre's_multiplicity_conjectures
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)
a 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
J-2_ring
ring A to be a normal ring. The criterion involves the following two conditions for A: R k : A p {\displaystyle R_{k}:A_{\mathfrak {p}}} is a regular
Serre's criterion for normality
Serre's_criterion_for_normality
ideal is a finite extension of a regular local ring. The Weierstrass preparation theorem can be used to show that the ring of convergent power series over
Weierstrass_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)
Type of integral domain
the formal power series ring R[[X]] over R is not a UFD. The Auslander–Buchsbaum theorem states that every regular local ring is a UFD. Z [ e 2 π i /
Unique_factorization_domain
Algebraic formula
local ring is regular if, and only if, it has finite global dimension. In turn this implies that the localization of a regular local ring is regular.
Auslander–Buchsbaum_formula
positivity conjecture that if R {\displaystyle R} is a commutative regular local ring, and P , Q {\displaystyle P,Q} are prime ideals of R {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Commutative algebra studies commutative rings, their ideals, and modules over such rings
theory) Integral closure Completion (ring theory) Formal power series Localization of a ring Local ring Regular local ring Localization of a module Valuation
List of commutative algebra topics
List_of_commutative_algebra_topics
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 below)
G-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
Hironaka_decomposition
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
American mathematician
algebras over a field). He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and, in collaboration
Maurice_Auslander
Algebraic theorem
commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by
Auslander–Buchsbaum_theorem
regular 1. A regular surface is one whose irregularity is zero. 2. Having no singularities; see regular local ring. 3. Symmetrical, as in regular polygon
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Study of objects of arithmetic interest over infinite towers of number fields
\Lambda =\mathbb {Z} _{p}[[\Gamma ]]} . This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this
Iwasawa_theory
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
In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal i {\displaystyle {\mathfrak {i}}}
Regular_ideal
which is the henselization of the ring of germs of rational functions. In particular, it is a regular local ring of dimension n. The global properties
Nash_function
Branch of algebraic geometry
Serre shows, it is not sufficient. The sum is finite, because the regular local ring O X , z {\displaystyle {\mathcal {O}}_{X,z}} has finite Tor-dimension
Intersection_theory
Construction of a ring of fractions
ring if and only if they are true for all its local rings. For example, a ring is regular if and only if all its local rings are regular local rings.
Localization (commutative algebra)
Localization_(commutative_algebra)
Branch of mathematics that studies algebraic structures
Neumann 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
American animated sitcom
Regular Show (known as Regular Show in Space during its eighth season) is an American animated sitcom created by J. G. Quintel for Cartoon Network. It
Regular_Show
Algebraic structure in ring theory
ISBN 978-3-11-016633-0, MR 1753146 Kunz, Ernst (1969), "Characterizations of regular local rings of characteristic p", American Journal of Mathematics, 91 (3): 772–784
Flat_module
American mathematician and historian (1942–present)
mathematics under Maurice Auslander with thesis The Brauer group of a regular local ring. He became at Federal City College an assistant professor and then
Victor_J._Katz
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
Algebra, a branch of mathematics
}}d_{1}={\begin{bmatrix}x+y&y\\x&-z\end{bmatrix}}} definition Given a regular local ring R {\displaystyle R} and an ideal I ⊂ R {\displaystyle I\subset R}
Matrix factorization (algebra)
Matrix_factorization_(algebra)
complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull: Any complete regular equicharacteristic
Cohen_structure_theorem
Subject area in mathematics
codimension 2 cycles on X. Inspired by this, Gersten conjectured that for a regular local ring R with fraction field F, Kn(R) injects into Kn(F) for all n. Soon
Algebraic_K-theory
These characters appear in the American animated television series Regular Show, created by J. G. Quintel for Cartoon Network. The series revolves around
List of Regular Show characters
List_of_Regular_Show_characters
Concept in commutative algebra
all geometrically regular so A is not a G-ring. It is a J-2 ring as all Noetherian local rings of dimension at most 1 are J-2 rings. It is also universally
Excellent_ring
geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme
Regular_scheme
German mathematician (1899–1971)
structure theorem Jacobson ring Local ring Prime ideal Real algebraic geometry Regular local ring Valuation ring Krull dimension Krull ring Krull topology Krull–Azumaya
Wolfgang_Krull
Type of ideal relevant for Noetherian rings
{grade}}(I)={\textrm {proj}}\dim(R/I).} A perfect ideal is unmixed. For a regular local ring R {\displaystyle R} a prime ideal I {\displaystyle I} is perfect if
Perfect_ideal
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
(algebra) Krull dimension Regular local ring Regular sequence Cohen–Macaulay ring Gorenstein ring Koszul complex Spectrum of a ring Zariski topology Kähler
List of algebraic geometry topics
List_of_algebraic_geometry_topics
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
Topics referred to by the same term
American auto racing team Range Life Records, an American record label Regular local ring Richard Lloyd Racing, a defunct British auto racing team RIG-I-like
RLR
Algebraic variety defined within an affine space
polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in more technical terms (see
Affine_variety
isomorphism. regular A regular scheme is a scheme where the local rings are regular local rings. For example, smooth varieties over a field are regular, while
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Set of a ring's prime ideals
identified with the affine scheme built over its ring of regular functions. The idea of the spectrum of a ring was introduced under that name by Alexander
Spectrum_of_a_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)
Topological structure in number theory
1 + T with a topological generator of G. This ring is a 2-dimensional complete Noetherian regular local ring, and in particular a unique factorization domain
Iwasawa_algebra
On polynomial rings over fields
Krull dimension. This result may be proven using Serre's theorem on regular local rings. Quillen–Suslin theorem Hilbert series and Hilbert polynomial D.
Hilbert's_syzygy_theorem
Direct sum of irreducible modules
semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is a von Neumann regular ring. Semisimple rings are of particular
Semisimple_module
Invariant of rings and modules
characterized using the notion of a regular sequence. Suppose that R {\displaystyle R} is a commutative Noetherian local ring with the maximal ideal m {\displaystyle
Depth_(ring_theory)
\operatorname {Spec} B} is regularly embedded into a regular scheme, then B is a complete intersection ring. The notion is used, for instance, in an essential
Regular_embedding
Argentine boxer (born 1983)
the WBA (Regular) super lightweight title from 2011 to 2012, and the WBA welterweight title from 2013 to 2014. A versatile brawler in the ring, Maidana
Marcos_Maidana
Topics referred to by the same term
neighborhoods of points Local ring, type of ring in commutative algebra Pub, a drinking establishment, known as a "local" to its regulars All pages with titles
Local
Concept in mathematics
studied in differential geometry. The ring of regular functions (that is the coordinate ring or more abstractly the ring of global sections of the structure
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
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
R\subseteq S} is a module-finite ring extension with R regular (here, R need not be local but the problem reduces at once to the local case), then R is a direct
Homological conjectures in commutative algebra
Homological_conjectures_in_commutative_algebra
American mathematician (1937–2024)
commutative algebra, particularly in the study of Noetherian local rings and graded rings. Judith Donovan was born to Dr. and Mrs. Edward J. Donovan in
Judith_D._Sally
Japanese mathematician (1930–1991)
articles. "Higher differential algebras of discrete valuation rings" is cited by "Regular local rings essentially of finite type over fields of prime characteristic"
Satoshi Suzuki (mathematician)
Satoshi_Suzuki_(mathematician)
semiprimitive ring or Jacobson semisimple ring is a ring whose Jacobson radical is zero. Von Neumann regular rings and primitive rings are semiprimitive
Glossary_of_ring_theory
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
Japanese boxer (born 1993)
(122 lbs) WBA (Regular) bantamweight champion (118 lbs) WBA (Unified) bantamweight champion (118 lbs) The Ring bantamweight champion (118 lbs) The Ring super bantamweight
Naoya_Inoue
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
Species of mammal from Madagascar
The ring-tailed lemur (Lemur catta) is a medium- to larger-sized strepsirrhine (wet-nosed) primate and the most internationally recognized lemur species
Ring-tailed_lemur
Algebraic structure with addition, multiplication, and division
distributes over addition. Even more succinctly: a field is a commutative ring in which 0 ≠ 1 and all nonzero elements are invertible under multiplication
Field_(mathematics)
Kazakhstani boxer (born 1982)
step into the ring to check his skills and he lost his first fight. Golovkin began boxing competitively in 1993, age 11, winning the local Karaganda Regional
Gennady_Golovkin
American professional wrestler (born 1983)
Eden became the regular ring announcer and backstage interviewer for SmackDown and Main Event. In late 2014, Eden made her pay-per-view ring announcing debut
Brandi_Rhodes
Latin letter A with overring
LATIN CAPITAL LETTER A WITH RING ABOVE U+00E5 å LATIN SMALL LETTER A WITH RING ABOVE Some type designers like using the "ring stick" state, as they think
Å
British professional boxer (1978–2025)
and one at welterweight. In 2005 he was named Fighter of the Year by The Ring magazine, the Boxing Writers Association of America, ESPN, and BoxingScene
Ricky_Hatton
Dwarf planet with a ring and two moons
2017, astronomers announced the discovery of a ring system around Haumea, representing the first ring system discovered for a trans-Neptunian object and
Haumea
British actor (born 1976)
Meriadoc "Merry" Brandybuck in Peter Jackson's film trilogy The Lord of the Rings (2001–2003), and Charlie Pace on the ABC television drama Lost (2004–2010)
Dominic_Monaghan
"Adjoints and polars of simple complete ideals in two-dimensional regular local rings" (PDF), Bulletin de la Société Mathématique de Belgique. Série A
Multiplier_ideal
Finger ring worn by clergy
An ecclesiastical ring is a finger ring worn by clergy, such as a bishop's ring. In Western Christianity, rings are worn by bishops of the Roman Catholic
Ecclesiastical_ring
American writer, actor and gamemaster (born 1988)
CollegeHumor) as a writer, performer, and producer. He is the creator and regular gamemaster for Dropout's actual play series Dimension 20 as well as the
Brennan_Lee_Mulligan
Concept in ring theory and homological algebra
in the above list. Serre proved that a commutative Noetherian local ring A is regular if and only if it has finite global dimension, in which case the
Global_dimension
Algebraic structure with "nice" duality properties
Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest
Frobenius_algebra
Ring in which every ideal is principal
direct product of principal Artinian local rings. A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure:
Principal_ideal_ring
Expressway ring road in Beijing, China
Daqing-Guangzhou Expressway. The Sixth Ring Road was opened for regular traffic on 12 September 2009. The 6th Ring Road runs within the confines of the
G4501 Beijing 6th Ring Expressway
G4501_Beijing_6th_Ring_Expressway
American businessman and sports team owner (born 1941)
give the ring as a gift and that his statement had been issued under pressure from the White House after Putin had kept the ring. The ring is on display
Robert_Kraft
Russian boxer (born 1990)
since April 2025, as well as the International Boxing Organization (IBO) and Ring magazine titles since February 2025. Previously, he held the World Boxing
Dmitry_Bivol
Module over a ring
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
Torsion-free_module
Equivalence class of objects sharing local properties at a point in a topological space
and for regular functions on an algebraic variety. The property that rings of germs are local rings is axiomatized by the theory of locally ringed spaces
Germ_(mathematics)
Nigerian-Greek basketball player (born 1994)
statistical categories and became the first player in NBA history to finish a regular season in the top 20 in all five statistics: total points, rebounds, assists
Giannis_Antetokounmpo
British-American period crime television drama
Nash (recurring, series 2; regular, series 3– ), another private detective Brian Bovell as Solomon (guest, series 2; regular, series 3), a shopkeeper specialising
Miss_Scarlet_and_The_Duke
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
American professional wrestler (born 1981)
Dynamite and is a part-time in-ring performer. He is also known for his tenure in WWE, where he performed under the ring name Daniel Bryan from 2010 to
Bryan_Danielson
Road in South Africa
The Pretoria Ring Road, also known as the Pretoria Bypass, is a collection of two bypasses that together form a partial ring road around the city of Pretoria
Pretoria_Ring_Road
REGULAR LOCAL-RING
REGULAR LOCAL-RING
Boy/Male
American, British, English
Loyal
Boy/Male
Gujarati, Haryanvi, Hindu, Indian, Kannada, Marathi, Telugu
Regular; Ethical; Good in Nature
Boy/Male
English American French
Faithful; unswerving.
Boy/Male
American, Australian, British, English, French
Faithful; True
Girl/Female
French
Loyal.
Boy/Male
Irish Welsh
Loyal.
Boy/Male
Italian Greek
Loyal.
Boy/Male
Indian
Loyal
Girl/Female
Indian
Loyal
Boy/Male
British, English
Loyal
Girl/Female
Hebrew
Precious.
Boy/Male
Hindu, Indian, Traditional
Conduct; Regular Performance of Worship
Boy/Male
Indian, Sanskrit
Connector; Regulator
Girl/Female
Muslim
Loyal
Boy/Male
Irish
Loyal.
Boy/Male
Irish
Loyal.
Boy/Male
English American
Loyal.
Boy/Male
Irish American Welsh
Loyal.
Boy/Male
Hindu, Indian, Tamil
Regular Winner
Boy/Male
Irish Welsh
Loyal.
REGULAR LOCAL-RING
REGULAR LOCAL-RING
Boy/Male
Arabic, Muslim
Allah's Messenger; Pure
Boy/Male
Indian
Perfection of religion (Islam)
Surname or Lastname
English (East Anglia)
English (East Anglia) : unexplained.Probably an Americanized spelling of German Rückert or Ruckert.
Girl/Female
French, German, Latin
Female Bear
Female
Celtic
, the white spirit, i.e. an angel.
Female
Chinese
beautiful wisdom.
Girl/Female
Tamil
Samridhhi | ஸமà¯à®°à¯€à®¤à¯à®¤à¯€
Good luck, Perfection, Wealth, Accomplishment, Welfare
Boy/Male
Bengali, Hindu, Indian
Sweet; Lovely
Girl/Female
Spanish
Sweet.
Boy/Male
British, Czechoslovakian, English
God of Love
REGULAR LOCAL-RING
REGULAR LOCAL-RING
REGULAR LOCAL-RING
REGULAR LOCAL-RING
REGULAR LOCAL-RING
a.
Of or pertaining to the jugular vein; as, the jugular foramen.
a.
Uttered or modulated by the voice; oral; as, vocal melody; vocal prayer.
a.
Constituted, selected, or conducted in conformity with established usages, rules, or discipline; duly authorized; permanently organized; as, a regular meeting; a regular physican; a regular nomination; regular troops.
a.
Having all the parts of the same kind alike in size and shape; as, a regular flower; a regular sea urchin.
v. t.
To cause to become regular; to regulate.
adv.
In a regular manner; in uniform order; methodically; in due order or time.
pl.
of Regulus
n.
A principle, practice, form of speech, or other thing of local use, or limited to a locality.
a.
Not regular; not bound by monastic vows or rules; not confined to a monastery, or subject to the rules of a religious community; as, a secular priest.
a.
Belonging to,or concerning, a focus; as, a focal point.
a.
Conformed to a rule; agreeable to an established rule, law, principle, or type, or to established customary forms; normal; symmetrical; as, a regular verse in poetry; a regular piece of music; a regular verb; regular practice of law or medicine; a regular building.
a.
Governed by rule or rules; steady or uniform in course, practice, or occurence; not subject to unexplained or irrational variation; returning at stated intervals; steadily pursued; orderlly; methodical; as, the regular succession of day and night; regular habits.
n.
One who is not regular; especially, a soldier not in regular service.
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.
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.
a.
Belonging to a monastic order or community; as, regular clergy, in distinction dfrom the secular clergy.
a.
Thorough; complete; unmitigated; as, a regular humbug.
pl.
of Tegula
n. pl.
A division of Echini which includes the circular, or regular, sea urchins.
a.
Not regular; not conforming to a law, method, or usage recognized as the general rule; not according to common form; not conformable to nature, to the rules of moral rectitude, or to established principles; not normal; unnatural; immethodical; unsymmetrical; erratic; no straight; not uniform; as, an irregular line; an irregular figure; an irregular verse; an irregular physician; an irregular proceeding; irregular motion; irregular conduct, etc. Cf. Regular.