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Regular Local Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal \mathfrak, and suppose a_1,\cdots,a_n is a minimal set of generators of \mathfrak. Then Krull's principal ideal theorem implies that n\geq\dim A, and A is regular whenever n=\dim A. The concept is motivated by its geometric meaning. A point x on an algebraic variety X is nonsingular (a smooth point) if and only if the local ring \mathcal_ of germs at x is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal ideal \mathfrak, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and p-adic number, ''p''-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. Several concepts of commutative algebras have been developed in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection ... [...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 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irvin Cohen
Irvin Sol Cohen (1917 – February 14, 1955) was an American mathematician at the Massachusetts Institute of Technology who worked on local rings. He was a student of Oscar Zariski at Johns Hopkins University. In his thesis he proved the Cohen structure theorem for complete Noetherian local rings. In 1946 he proved the unmixedness theorem for power series rings. As a result, Cohen–Macaulay rings are named after him and Francis Sowerby Macaulay. Cohen and Abraham Seidenberg published their Cohen–Seidenberg theorems, also known as the going-up and going-down theorems. He also coauthored articles with Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St And .... One of his doctoral students was R. Duncan Luce. Death Cohen died unexpectedly in 1955 one week after ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen Structure Theorem
In mathematics, the Cohen structure theorem, introduced by , describes the structure of complete Noetherian 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 ...s. Some consequences of Cohen's structure theorem include three conjectures of Krull: *Any complete regular equicharacteristic Noetherian local ring is a ring of formal power series over a field. (Equicharacteristic means that the local ring and its residue field have the same characteristic, and is equivalent to the local ring containing a field.) *Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension. *Any complete Noetherian local ring is the image of a complete regular Noetherian local ring. Statement The mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring And A Module
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''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 this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Integer
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinity (mathematics), infinite) decimal representation, decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in Ternary numeral system, base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\[5mu] \tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series (mathematics), series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # ''R'' is a local ring, a principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local ring, a Dedekind domain, and not a field. # ''R'' is Noetherian and a local domain whose unique maximal ideal is principal, and not a field. # ''R'' is integrally closed, Noetherian, and a local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element (up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring ''R'' form an ideal of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring ''R''/''I'' is reduced if and only if ''I'' is a radical ideal. Let \mathcal_R denote nilradical of a commutative ring R. There is a functor R \mapsto R/\mathcal_R of the category of commutative rings \text into the category of reduced rings \text and it is left adjoint to the inclusion functor I of \text into ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Ring Homomorphism
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module ''M'' over a ring ''R'' is ''flat'' if taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. Definition A left module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of right -modules, the map : \varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
