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Dimension Theory (algebra)
In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a ''theory'' for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes 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 is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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. Formally, 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 n such that I_=I_=\cdots. Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively 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 the Noetherian property ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Avoidance
In algebra, the prime avoidance lemma says that if an ideal ''I'' in a commutative ring ''R'' is contained in a union of finitely many prime ideals ''P''''i'''s, then it is contained in ''P''''i'' for some ''i''. There are many variations of the lemma (cf. Hochster); for example, if the ring ''R'' contains an infinite field or a finite field of sufficiently large cardinality, then the statement follows from a fact in linear algebra that a vector space over an infinite field or a finite field of large cardinality is not a finite union of its proper vector subspaces. Statement and proof The following statement and argument are perhaps the most standard. Statement: Let ''E'' be a subset of ''R'' that is an additive subgroup of ''R'' and is multiplicatively closed. Let I_1, I_2, \dots, I_n, n \ge 1 be ideals such that I_i are prime ideals for i \ge 3. If ''E'' is not contained in any of I_i's, then ''E'' is not contained in the union \cup I_i. Proof by induction on ''n'': The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Samuel Function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203. of a nonzero finitely generated module M over a commutative Noetherian local ring A and a primary ideal I of A is the map \chi_^:\mathbb\rightarrow\mathbb such that, for all n\in\mathbb, :\chi_^(n)=\ell(M/I^M) where \ell denotes the length over A. It is related to the Hilbert function of the associated graded module \operatorname_I(M) by the identity : \chi_M^I (n)=\sum_^n H(\operatorname_I(M),i). For sufficiently large n, it coincides with a polynomial function of degree equal to \dim(\operatorname_I(M)), often called the Hilbert-Samuel polynomial (or Hilbert polynomial).Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969. Examples For the ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin–Rees Lemma
In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive characterization of the lemma involves the notion that a submodule ''N'' of a module ''M'' over some ring ''A'' with specified ideal ''I'' holds ''a priori'' two topologies: one induced by the topology on ''M,'' and the other when considered with the ''I-adic'' topology over ''A.'' Then Artin-Rees dictates that these topologies actually coincide, at least when ''A'' is Noetherian and ''M'' finitely-generated. One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves. Statement Let ''I'' be an ideal in a Noetherian r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length Of A Module
In algebra, the length of a module over a ring R is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules. For vector spaces (modules over a field), the length equals the dimension. If R is an algebra over a field k, the length of a module is at most its dimension as a k-vector space. In commutative algebra and algebraic geometry, a module over a Noetherian commutative ring R can have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules have infinite length. Modules of finite length are Artinian modules and are fundamental to the theory of Artinian rings. The degree of an algebraic variety inside an affine or projective space is the length of the coordinate ring of the zero-dimensional intersection of the variety with a generic linear subspace of complementary dimension. More gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Graded Ring
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Associated may refer to: *Associated, former name of Avon, Contra Costa County, California *Associated Hebrew Schools of Toronto, a school in Canada *Associated Newspapers, former name of DMG Media, a British publishing company See also *Association (other) *Associate (other) Associate may refer to: Academics * Associate degree, a two-year educational degree in the United States, and some areas of Canada * Associate professor, an academic rank at a college or university * Technical associate or Senmonshi, a Japa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Poincaré Series
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say t, where the coefficient of t^n gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n. It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n'' > 0. For example, in the ring of integers Z, (''p''''n'') is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. Examples and properties * The definition can be rephrased in a more symmetric m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non- units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
