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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
work of Oscar Zariski a few years later, who showed that geometrically, a regular local ring corresponds to a smooth point on an algebraic variety. Let
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
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
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
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
Concept in commutative algebra
localization rings of these rings. A ring R containing a field k is called geometrically regular over k if for any finite extension K of k the ring R⊗kK is
Excellent_ring
Type of ring in commutative algebra
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
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
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
Construction within abstract algebra
prime ideals, 𝔭1,…,𝔭r (e.g., a Noetherian reduced ring). Then Q(A)≃∏i=1rQ(A/𝔭i). Geometrically, Spec(Q(A)) is the Artinian scheme consisting (as a
Total_ring_of_fractions
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
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
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
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 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
This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in
Quasiregular_element
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
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
inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian
Catenary_ring
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
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
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)
Set of a ring's prime ideals
algebraic variety can be identified with the affine scheme built over its ring of regular functions. For the related, more general notion of prime and maximal
Spectrum_of_a_ring
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
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
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
On the effects of changing the ring of ''K''-groups
from a ring to a polynomial ring or a Laurent polynomial ring. The theorem establishes homotopy invariance for the K-theory of suitable regular rings and
Fundamental theorem of algebraic K-theory
Fundamental_theorem_of_algebraic_K-theory
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)
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)
Commutative algebra theorem
projective module over a polynomial ring is free. Geometrically, finitely generated projective modules over the ring R[x1,…,xn] correspond to vector bundles
Quillen–Suslin_theorem
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)
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
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
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)
Three-dimensional geometric shape
discovered a geometric net consisting of sixty isosceles triangles that, when scored and folded, form a movable three-dimensional ring capable of continuous
Kaleidocycle
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
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
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)
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)
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
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
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
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
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
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
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
Algebraic structure designed for geometry
form of the geometric product. The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, a and b
Geometric_algebra
Branch of algebraic geometry
was given by Serre: Let the ambient variety X be smooth (or all local rings regular). Further let V and W be two (irreducible reduced closed) subvarieties
Intersection_theory
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
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)
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
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)
Algebraic ring that need not have additive negative elements
a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse
Semiring
Number with a real and an imaginary part
be performed as 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
(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
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
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
Positive real number which when multiplied by itself gives 5
tables. Since 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
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
Solid with eight equal triangular faces
more generally, a regular polyhedron. If the faces are isosceles triangles, the regular octahedron becomes a square bipyramid. The regular octahedron is an
Regular_octahedron
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
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
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
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
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
Shape with three equal sides
sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle. It is a special case of an isosceles triangle
Equilateral_triangle
Finite extension of the rationals
many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which are the function
Algebraic_number_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
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
Multi-dimensional generalization of triangle
Δn(R)=Spec(R[Δn]) with R[Δn]:=R[x1,…,xn+1]/(1−∑xi) the ring of regular functions on the algebraic n-simplex (for any ring R). By using the same definitions as for the
Simplex
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
Line 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
Mathematical object studied in the field of algebraic geometry
schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such
Algebraic_variety
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
rhombicuboctahedron 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-octahedral_honeycomb
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
Measure of a mathematical object studied in the field of algebraic geometry
an affine or projective space. The maximal length of a regular sequence in the coordinate ring A. This the algebraic translation of the preceding definition
Dimension of an algebraic variety
Dimension_of_an_algebraic_variety
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
rhombicuboctahedron 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
Tetrahedral-square tiling honeycomb
Tetrahedral-square_tiling_honeycomb
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
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
Only regular space-filling tessellation of the cube
a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille. A geometric honeycomb
Cubic_honeycomb
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
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
Algebraic structure in ring theory
Generic flatness Flat morphism von Neumann regular ring – rings over which all modules are flat. Normally flat ring Proof: Suppose f:R→S is faithfully flat
Flat_module
Regular object in four dimensional geometry
three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern. The regular complex
24-cell
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
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
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) of
Normal_scheme
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
Branch of number theory
algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization
Algebraic_number_theory
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
Term in algebraic geometry
where X → Z {\displaystyle X\to Z} is proper, surjective, and has geometrically connected fibers, and Z → Y {\displaystyle Z\to Y} is finite. Chow's
Proper_morphism
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
Shape with four equal sides and angles
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles
Square
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