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Approximation Property (ring Theory)
In algebra, a commutative Noetherian ring ''A'' is said to have the approximation property with respect to an ideal ''I'' if each finite system of polynomial equations with coefficients in ''A'' has a solution in ''A'' if and only if it has a solution in the ''I''-adic completion of ''A''. The notion of the approximation property is due to Michael Artin. See also *Artin approximation theorem *Popescu's theorem In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and ... Notes References * * * * Ring theory {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
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 only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completion (ring Theory)
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions ''R'' on a space ''X'' concentrates on a formal neighborhood of a point of ''X'': heuristically, this is a neighborhood so small that ''all'' Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when ''R'' has a metric given by a non-Archimedean absolute value. General construction Suppose that ''E'' is an abelian group with a descending filtration : E = F^0 E \supset F^1 E \supset F^2 E \supset \cdots \, of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Artin
Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Faculty profile , MIT mathematics department, retrieved 2011-01-03 Life and career Michael Artin or Artinian was born in , Germany, and brought up in . His parents were Natalia Naumovna Jasny (Natascha) and[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Approximation Theorem
In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field ''k'' are well-approximated by the algebraic functions on ''k''. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = \Complex); and an algebraic version of this theorem in 1969. Statement of the theorem Let \mathbf = x_1, \dots, x_n denote a collection of ''n'' indeterminates, k \mathbf the ring of formal power series with indeterminates \mathbf over a field ''k'', and \mathbf = y_1, \dots, y_n a different set of indeterminates. Let :f(\mathbf, \mathbf) = 0 be a system of polynomial equations in k mathbf, \mathbf/math>, and ''c'' a positive integer. Then given a formal power series solution \hat(\mathbf) \in k \mathbf, there is an algebraic solution \mathbf(\mathbf) consisting of algebraic functions (more precisely, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Popescu's Theorem
In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and only if ''B'' is a direct limit of smooth ''A''-algebras. For example, if ''A'' is a local G-ring (e.g., a local excellent ring) and ''B'' its completion, then the map ''A'' → ''B'' is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by Richard Swan. The usual proof of the Artin approximation theorem In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field (mathematics), field ''k'' are well-approximated by the algebraic functions on ''k' ... relies crucially on Popescu's theorem. Popescu's result was proved by an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Columbia University
Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhattan, Columbia is the oldest institution of higher education in New York and the fifth-oldest institution of higher learning in the United States. It is one of nine colonial colleges founded prior to the Declaration of Independence. It is a member of the Ivy League. Columbia is ranked among the top universities in the world. Columbia was established by royal charter under George II of Great Britain. It was renamed Columbia College in 1784 following the American Revolution, and in 1787 was placed under a private board of trustees headed by former students Alexander Hamilton and John Jay. In 1896, the campus was moved to its current location in Morningside Heights and renamed Columbia University. Columbia scientists and scholars have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagoya Mathematical Journal
is the largest city in the Chūbu region, the fourth-most populous city and third most populous urban area in Japan, with a population of 2.3million in 2020. Located on the Pacific coast in central Honshu, it is the capital and the most populous city of Aichi Prefecture, and is one of Japan's major ports along with those of Tokyo, Osaka, Kobe, Yokohama, and Chiba. It is the principal city of the Chūkyō metropolitan area, which is the third-most populous metropolitan area in Japan with a population of 10.11million in 2020. In 1610, the warlord Tokugawa Ieyasu, a retainer of Oda Nobunaga, moved the capital of Owari Province from Kiyosu to Nagoya. This period saw the renovation of Nagoya Castle. The arrival of the 20th century brought a convergence of economic factors that fueled rapid growth in Nagoya, during the Meiji Restoration, and became a major industrial hub for Japan. The traditional manufactures of timepieces, bicycles, and sewing machines were followed by the produ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
