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...

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...

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...

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...

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...

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...

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...

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...

a strong correspondence between algebraic sets and ideals of polynomial rings. This led to a parallel development of algebraic geometry, and its algebraic...

algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series...

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....

integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division...

mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra...

is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring R are also called the regular functions or the polynomial...

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]...

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...

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...

names recall often their geometric origin; for example "Krull dimension", "localization of a ring", "local ring", "regular ring". An affine algebraic variety...

commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R {\displaystyle R} is the set of all prime ideals of R {\displaystyle R}...

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 ⁡ (...

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...

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...

ring Local ring Noetherian and artinian rings Ordered ring Poisson ring Reduced ring Regular ring Ring of periods SBI ring Valuation ring and discrete...

inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian...

projective module over a polynomial ring is free. Geometrically, finitely generated projective modules over the ring R [ x 1 , … , x n ] {\displaystyle...

a geometrically regular local ring. acceptable ring Acceptable rings are generalizations of excellent rings, with the conditions about regular rings in...

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 made...

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...

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...

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...