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Glossary Of Commutative Algebra
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative with identity 1. !$@ A B C D E F G H . I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Algebraic Geometry Topics
This is a list of algebraic geometry topics, by Wikipedia page. Classical topics in projective geometry * Affine space *Projective space *Projective line, cross-ratio *Projective plane **Line at infinity **Complex projective plane *Complex projective space *Plane at infinity, hyperplane at infinity *Projective frame *Projective transformation *Fundamental theorem of projective geometry *Duality (projective geometry) *Real projective plane *Real projective space *Segre embedding of a product of projective spaces *Rational normal curve Algebraic curves *Conics, Pascal's theorem, Brianchon's theorem * Twisted cubic *Elliptic curve, cubic curve **Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions **Elliptic integral **Complex multiplication **Weil pairing *Hyperelliptic curve *Klein quartic *Modular curve ** Modular equation **Modular function **Modular group ** Supersingular primes *Fermat curve * Bézout's theorem * Brill–Noether theory *Genus (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrically Regular Local Ring
In algebraic geometry, a 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 schemes are defined in a similar way. In older terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points. Over fields that are of characteristic 0, or algebraically closed, or more generally perfect, geometrically regular rings are the same as regular rings. Geometric regularity originated when Claude Chevalley and André Weil pointed out to that, over non-perfect fields, the Jacobian criterion for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular. A Noetherian local ring containing a field ''k'' is geometrically regular over ''k'' if and only if it is formally smooth over ''k''. Examples gave the following two examples of local rings tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra With Straightening Law
In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring ''R'', together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian. Hodge algebras were introduced by , who named them after W. V. D. Hodge Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer. His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now c .... References * Commutative algebra {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin. In the presence of the axiom of choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead. Like Noetherian modules, Artinian modules enjoy the following heredity property: * If ''M'' is an Artinian ''R''-module, then so is any submodule and any quotient of ''M''. The converse also holds: * If ''M'' is any ''R''-module and ''N'' any Artinian submodule such that ''M''/''N'' is Artinian, then ''M'' is Artinian. As a consequence, any finitely-generated module over an Artinian ring is Artinian.Lam (2001), Proposition 1.21, p. 19 Since an Artinian ring ... [...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 Hamburg, Germany, and brought up in Indiana. His parents were Natalia Naumovna Jasny (Natascha) and Emil Artin, preeminent algebraist ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''-module. Choose a non-empty subset ''S'' of ''M''. The annihilator of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modification of "" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Irreducible
In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. proved that if a local ring of an algebraic variety is a normal ring In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ..., then it is analytically irreducible. There are many examples of reduced and irreducible local rings that are analytically reducible, such as the local ring of a node of an irreducible curve, but it is hard to find examples that are also normal. gave such an example of a normal Noetherian local ring that is analytically reducible. Nagata's example Suppose that ''K'' is a field of characteristic not 2, and ''K'' is the formal power series ring over ''K'' in 2 variables. Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Unramified
In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent). The following rings are analytically unramified: * pseudo-geometric reduced ring. * excellent reduced ring. showed that every local ring of an algebraic variety is analytically unramified. gave an example of an analytically ramified reduced local ring. Krull showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more precisely he showed that a 1-dimensional normal Noetherian local domain is analytically unramified if and only if its integral closure is a finite module. This prompted to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However gave an example of a 2-dimensional normal analytically ramified Noetherian local ring. Nagata also showed that a slightly stronger version of Zariski's question is correct: if the normalization of every finite extensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Normal
In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field. proved that if a local ring of an algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ... is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem. gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal. References * * * * * Commutative algebra {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |