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geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular
Geometrically_regular_ring
Type of ring in commutative algebra
domains is regular, but not an integral domain. Geometrically regular ring quasi-free ring A local von Neumann regular ring is a division ring, so the two
Regular_local_ring
Rings admitting weak inverses
V-ring. R has the right-lifting property against the ring homomorphism Z[t] → Z[t±] × Z determined by t ↦ (t, 0), or said geometrically, every regular function
Von_Neumann_regular_ring
perfect field is smooth. For an example of a regular scheme that is not smooth, see Geometrically regular ring § Examples. Étale morphism Dimension of an
Regular_scheme
excellent ring. A (Noetherian) ring R containing a field k is called geometrically regular over k if for any finite extension K of k the ring R ⊗k K is
G-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
Concept in commutative algebra
are not 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
Excellent_ring
Branch of mathematics
a strong correspondence between algebraic sets and ideals of polynomial rings. This led to a parallel development of algebraic geometry, and its algebraic
Geometry
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
is 0.7–0.9. During a ring plane-crossing event in 2007 the γ ring disappeared, which means it is geometrically thin like the ε ring and devoid of dust.
Rings_of_Uranus
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
Type of commutative ring in mathematics
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality
Cohen–Macaulay_ring
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
Branch of mathematics
identified with the ring of polynomial functions in n variables over k. Therefore, the set of the regular functions on An is a ring, which is denoted k[An]
Algebraic_geometry
Branch of algebra that studies commutative rings
names recall often their geometric origin; for example "Krull dimension", "localization of a ring", "local ring", "regular ring". An affine algebraic variety
Commutative_algebra
Three linked but pairwise separated rings
two of them are bound. Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked
Borromean_rings
Algebraic structure
algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series
Polynomial_ring
Construction within abstract algebra
Noetherian reduced ring). Then Q ( A ) ≃ ∏ i = 1 r Q ( A / p i ) . {\displaystyle Q(A)\simeq \prod _{i=1}^{r}Q(A/{\mathfrak {p}}_{i}).} Geometrically, Spec (
Total_ring_of_fractions
a geometrically regular local ring. acceptable ring Acceptable rings are generalizations of excellent rings, with the conditions about regular rings in
Glossary of commutative algebra
Glossary_of_commutative_algebra
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
Local ring in commutative algebra
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, as
Gorenstein_ring
Algebraic structure with addition, multiplication, and division
type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the plane, with Cartesian coordinates given
Field_(mathematics)
that is locally of finite type and regular over k. 3. A smooth scheme over a field k is a scheme X that is geometrically smooth: X × k k ¯ {\displaystyle
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Commutative algebra theorem
projective module over a polynomial ring is free. Geometrically, finitely generated projective modules over the ring R [ x 1 , … , x n ] {\displaystyle
Quillen–Suslin_theorem
Branch of mathematics
y, and the resulting quotient ring is the polynomial ring in two variables, C[x, y]. Geometrically, the polynomial ring in two variables represents the
Noncommutative algebraic geometry
Noncommutative_algebraic_geometry
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
Geometric objects with a common centre
including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics. Geometric objects are coaxial if they
Concentric_objects
Generalization of algebraic variety
that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety
Scheme_(mathematics)
Chemical compound
it is not possible geometrically for all the angles and bond lengths to be equal except if it is in the form of a flat regular pentagon. Envelope 3D
Cyclopentane
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)
Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a
Weierstrass_ring
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)
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
Concept in algebraic geometry
\oplus k[t]\cdot x^{n-1}} as k [ t ] {\displaystyle k[t]} -modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of
Finite_morphism
Term in algebraic geometry
case: the ring of regular functions on a proper scheme X over a field k has finite dimension as a k-vector space. By contrast, the ring of regular functions
Proper_morphism
Generalization of vector spaces from fields to rings
module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies
Module_(mathematics)
Geometric pattern characteristic of Muslim art
diminishing in size as they rise. They are often elaborately decorated. Geometrically patterned stained glass is used in a variety of settings in Islamic
Islamic_geometric_patterns
Natural satellites of the planet Neptune
Neptune's only regular satellites, all with prograde orbits close to the planet's equatorial plane; some orbit among Neptune's rings. Including the largest
Moons_of_Neptune
realization of this 1-polytope is regular. It has the Schläfli symbol { }, or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a
List_of_regular_polytopes
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
Concept in algebraic geometry
finitely many fibers of f {\displaystyle f} are geometrically integral and all fibers are geometrically connected (by Zariski's connectedness theorem)
Canonical_bundle
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
Solid with eight equal triangular faces
p. 141. ISBN 978-0-486-83654-6. Sibley, Thomas Q. (2015). Thinking Geometrically: A Survey of Geometries. Mathematical Association of American. p. 53
Regular_octahedron
Compact Riemann surface of genus 3
tiling is topologically but not geometrically the 3 4 | 4 tiling). This immersion can also be used to geometrically construct the Mathieu group M24 by
Klein_quartic
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
Branch of algebraic geometry
Poincaré duality, it turns out that there is a way to think of this geometrically. If possible, choose representative n-dimensional submanifolds A, B
Intersection_theory
inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian
Catenary_ring
Property of objects which appear unchanged after a partial rotation
which are geometrically different, see dihedral symmetry groups in 3D. 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron
Rotational_symmetry
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)
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
Semiprimitive_ring
Number with a real and an imaginary part
{\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.} The addition can be geometrically visualized as follows: the sum of two complex numbers a and b, interpreted
Complex_number
Construction of a ring of fractions
localization originated in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study
Localization (commutative algebra)
Localization_(commutative_algebra)
Natural satellites of the planet Saturn
regular moons orbit near the edges of or within gaps in the main rings, some of which act as shepherd moons of the dense A Ring and the narrow F Ring
Moons_of_Saturn
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
Pictorial representation of symmetry
3). And E8 folds into 2 copies of H4, the second copy scaled by τ. Geometrically this corresponds to orthogonal projections of uniform polytopes and
Coxeter–Dynkin_diagram
Way to divide polygon into smaller parts
smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over
Finite_subdivision_rule
Lets one glue 2 sheaves over an infinitesimal neighborhood of an algebraic curve point
we have a ring A and an element f, and two modules: an Af-module F and an Â-module G, together with an isomorphism φ as above. Geometrically, we are given
Beauville–Laszlo_theorem
2022 television season
first season of the American fantasy television series The Lord of the Rings: The Rings of Power is based on J. R. R. Tolkien's history of Middle-earth, primarily
The Lord of the Rings: The Rings of Power season 1
The_Lord_of_the_Rings:_The_Rings_of_Power_season_1
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
Topology on prime ideals and algebraic varieties
ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology
Zariski_topology
Doughnut-shaped surface of revolution
coplanar with the circle. The main types of tori include ring tori, horn tori, and spindle tori. A ring torus is sometimes colloquially referred to as a doughnut
Torus
Discrete valuation field
wants to take into account. Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes
Higher_local_field
Branch of mathematics
the value of other variables. Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph. To do so, the different
Algebra
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
Positive real number which when multiplied by itself gives 5
{\sqrt {5}}} is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of
Square_root_of_5
(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
Polytope in 8-dimensional geometry
called it an 8-ic semi-regular figure. Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node
4_21_polytope
Region between two concentric circles
is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival
Annulus_(mathematics)
Concept in abstract algebra
valuation ring. This is useful for building intuition with the valuative criterion of properness. For an example more geometrical in nature, take the ring R =
Discrete_valuation_ring
tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells. A geometric honeycomb is a space-filling of polyhedral
Cubic-triangular tiling honeycomb
Cubic-triangular_tiling_honeycomb
then the geometric series ∑ 0 ∞ x n {\displaystyle \sum _{0}^{\infty }x^{n}} converges. Consequently, every such x is quasiregular. If R is a ring and S
Quasiregular_element
Mathematical element
z]/(xy)} is the ring C [ x , z ] × C [ y , z ] {\displaystyle \mathbb {C} [x,z]\times \mathbb {C} [y,z]} since geometrically, the first ring corresponds to
Integral_element
Segment in a circle or sphere from its center to its perimeter or surface
The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow
Radius
Geometric operation
polytope. When omnitruncation is applied to a regular polytope (or honeycomb) it can be described geometrically as a Wythoff construction that creates a maximum
Omnitruncation
Regular object in four dimensional geometry
5-cell, but none of the pentagonal polytopes. The geometric relationships among all of these regular polytopes can be observed in a single 24-cell or the
24-cell
Mathematical concept
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication
Opposite_ring
Uniform 6-polytope
defined by all permutations of rings in this Coxeter-Dynkin diagram: . The 122 is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin
1_22_polytope
Non-orientable surface with one edge
every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral surface. To be realizable, it is necessary and sufficient
Möbius_strip
Seven-dimensional geometric object
by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the regular polytopes
Uniform_7-polytope
Scheme theory concept
Geometrically regular. Geometrically normal. If in addition f is proper, then the same is true for each of the following properties: Geometrically reduced
Flat_morphism
Moment of inertia of diff geometric shapes
McGraw-Hill. p. 911. ISBN 0-07-004389-2. Eric W. Weisstein. "Moment of Inertia — Ring". Wolfram Research. Retrieved 2016-12-14. Jeremy Tatum (14 April 2017). "2
List_of_moments_of_inertia
Construction in homological algebra
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 about the
Koszul_complex
Concept in algebra
is a regular point on the curve; i.e., the local ring R at the point is a regular local ring of Krull dimension one or a discrete valuation ring. For
Valuation_ring
Set of principles for modeling solid geometry
areas of geometric modeling and computer graphics, such as 3D modeling, by its emphasis on physical fidelity. Together, the principles of geometric and solid
Solid_modeling
Multi-dimensional generalization of triangle
x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.} the ring of regular functions on the algebraic n-simplex (for any ring R {\displaystyle R} ). By using the same definitions
Simplex
Mathematical object studied in the field of algebraic geometry
Y at a point of Y may be non-reduced, even if X and Y are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal"
Algebraic_variety
Class of 4-dimensional polytopes
6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms
Uniform_4-polytope
Concept in algebraic geometry
Equivalently, the ideal in the polynomial ring generated by all gi and all those minors is the whole polynomial ring. In geometric terms, the matrix of derivatives
Smooth_scheme
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
Three-dimensional geometric shape
continuously twisted around a ring axis, showing 4 sets of 6 triangular faces. The kaleidocycle is invariant under twists about its ring axis by k π / 2 {\displaystyle
Kaleidocycle
Covering by shapes without overlaps or gaps
sometimes displaying geometric patterns. In 1619, Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations
Tessellation
Module over a sheaf of differential operators
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of
D-module
Uniform 6-dimensional polytope
polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic
Uniform_6-polytope
Uniform 6-polytope
its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected
2_21_polytope
Shape with four equal sides and angles
The decision to go oxymoron with a squared "ring" had taken place by the late 1830s ... Despite the geometric shift, the language was set. Sciarappa, Luke;
Square
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)
Colored cement bands in sedimentary rocks
arranged in a regular repeating pattern. Liesegang rings are distinguishable from other sedimentary structures by their concentric or ring-like appearance
Liesegang_rings_(geology)
Algebra with unique prime factorization
other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic
Dedekind_domain
Mathematical concept
understanding of different types of rings and domains. In this article, all rings are commutative rings, and ring and overring share the same identity
Overring
Type of geometrical object
Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: Selected regular and uniform 10-polytopes from each family include:
Uniform_10-polytope
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
Girl/Female
Hebrew
Precious.
Male
Scandinavian
Scandinavian form of German Reginar, RAGNAR means "wise warrior."
Male
Spanish
Spanish form of Roman Latin Regulus, RÉGULO means "ruler."
Boy/Male
Hindu, Indian, Tamil
Regular Winner
Surname or Lastname
English, of Welsh origin
English, of Welsh origin : variant of Bowen, with the addition of the regular English patronymic suffix -s.Altered spelling of Dutch Bouwens, a variant of Bauwens.
Boy/Male
Shakespearean
King Henry IV, Part 1 and 2' An irregular humorist.
Surname or Lastname
English, of Welsh origin
English, of Welsh origin : variant of Bevan, with the addition of the regular English patronymic suffix -s.
Boy/Male
Gujarati, Haryanvi, Hindu, Indian, Kannada, Marathi, Telugu
Regular; Ethical; Good in Nature
Male
German
A derivative of German Reginar, RAINER means "wise warrior."
Boy/Male
Shakespearean
King Henry IV, Part 1 and 2' Edward Poins, an irregular humorist.
Surname or Lastname
English (Devon)
English (Devon) : unexplained. Possibly an irregular variant of Birchall.
Surname or Lastname
North German
North German : variant of Asch.English : variant spelling of Ash (asche was the regular Middle English spelling of this word).
Girl/Female
Muslim/Islamic
One who remembers Allah regularly
Girl/Female
Muslim
One who remembers Allah regularly
Surname or Lastname
English
English : nickname probably for a tenant whose feudal obligations included a regular payment in cash or kind (for example bread or salt) of a halfpenny.
Boy/Male
Hindu, Indian, Traditional
Conduct; Regular Performance of Worship
Male
Italian
Italian form of German Reginar, RANIERO means "wise warrior."
Girl/Female
Indian
One who remembers Allah regularly
Boy/Male
Indian, Sanskrit
Connector; Regulator
Girl/Female
Arabic, Muslim
Pilgrimage to Makkah Other than Regular Hajj Days
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
Biblical
same as Joshua
Boy/Male
Christian, German, Swedish
The Father of Peace; God; Oak Meadow; My Father is Peace
Boy/Male
American, British, English
Roofer
Boy/Male
American, Anglo, Australian, British, Chinese, English, Jamaican
Brock's Town; Bracc's Settlement
Girl/Female
Assamese, Bihari, Gujarati, Hindu, Indian, Telugu
Prosperity; To be Successful; To Succeed; To Grow; To Increase; To Make Gain
Boy/Male
American, Anglo, British, Christian, English
Nobleman; Chief; Leader; Prince; Warrior
Girl/Female
Indian
Well spoken.
Girl/Female
Hindu, Indian, Marathi
Rare
Boy/Male
Greek
Earth-lover. Of Demeter. Demeter is the mythological Greek goddess of corn and harvest. She...
Girl/Female
Indian
Fragrant, Sweet smelling, Another name for Paarvati
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
GEOMETRICALLY REGULAR-RING
a.
Thorough; complete; unmitigated; as, a regular humbug.
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.
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.
a.
Having all the parts of the same kind alike in size and shape; as, a regular flower; a regular sea urchin.
a.
Belonging to a monastic order or community; as, regular clergy, in distinction dfrom the secular clergy.
n. pl.
A division of Echini which includes the circular, or regular, sea urchins.
a.
Fig.: Lean; lank; raw-boned; ungraceful; sharp and stiff in character; as, remarkably angular in his habits and appearance; an angular female.
a.
Of or pertaining to a tile; resembling a tile, or arranged like tiles; consisting of tiles; as, a tegular pavement.
v. t.
To cause to become regular; to regulate.
pl.
of Regulus
n.
A secular ecclesiastic, or one not bound by monastic rules.
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.
Pertaining to, or according to the rules or principles of, geometry; determined by geometry; as, a geometrical solution of a problem.
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.
n.
One who is not regular; especially, a soldier not in regular service.
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.
adv.
In a regular manner; in uniform order; methodically; in due order or time.
a.
Measured by an angle; as, angular distance.
pl.
of Tegula
a.
Of or pertaining to the jugular vein; as, the jugular foramen.