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Integral Closure Of An Ideal
In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by \overline, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist a_i \in I^i such that :r^n + a_1 r^ + \cdots + a_ r + a_n = 0. It is similar to the integral closure of a subring. For example, if ''R'' is a domain, an element ''r'' in ''R'' belongs to \overline if and only if there is a finitely generated ''R''-module ''M'', annihilated only by zero, such that r M \subset I M. It follows that \overline is an ideal of ''R'' (in fact, the integral closure of an ideal is always an ideal; see below.) ''I'' is said to be integrally closed if I = \overline. The integral closure of an ideal appears in a theorem of Rees that characterizes an analytically unramified ring. Examples *In \mathbb, y/math>, x^i y^ is integral over (x^d, y^d). It satisfies the equation r^ + (-x^ y^) = 0, where a_d=-x^y^ is in the ideal. *Radical ideals (e.g., prime ideals) are integrally clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and " algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field ''k'' of the rationals Q form a subring of ''k'', called the ring of integers of ''k'', a central object of study in algebraic number theory. In this article, the term '' ring'' will be understood to mean ''commutative ring'' with a multiplicative identity. Definition Let B be a ring and let A \subset B be a subring of B. An el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Rees (mathematician)
David Rees FRS (29 May 1918 – 16 August 2013) was a British professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter from 1958 to 1983. During the Second World War, Rees was active on Enigma research in Hut 6 at Bletchley Park. Early life Rees was born in Abergavenny to David Rees (1881–), a corn merchant, and his wife Florence Gertrude (Gertie) née Powell (1884–1970), the 4th out of 5 children. Despite periods of ill health and absence, he successfully completed his early education at King Henry VIII Grammar School. Education and career Rees won a scholarship to Sidney Sussex College, Cambridge, supervised by Gordon Welchman and graduating in summer 1939. On completion of his education, he initially worked on semigroup theory; the Rees factor semigroup is named after him. He also characterised completely simple and completely 0-simple semigroups, in what is nowadays known as Rees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Unramified Ring
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 extension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Ideal
Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics, the political intent of fundamental societal change * Radical Party (other), several political parties *Radicals (UK), a British and Irish grouping in the early to mid-19th century *Radicalization *Politicians from the Radical Civic Union Ideologies * Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes *Radical feminism, a perspective within feminism that focuses on patriarchy * Radical Islam, or Islamic extremism * Radical Christianity * Radical veganism, a radical interpretation of veganism, usually combined with anarchism *Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation Science and mathematics Science *Radical (chemistry), an atom, molecule, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' that is a root of a monic polynomial with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed, as shown by the following chain of class inclusions: An 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 prime ideals are integrally closed domains is a normal ring. Basic properties Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a field extension of ''K''. Then ''x''∈''L'' is integral over ''A'' if and only if it is algebraic over ''K'' and its minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. Definitions and properties Let \mathbb be a field and R = \mathbb /math> be the polynomial ring over \mathbb with ''n'' indeterminates x = x_1, x_2, \dotsc, x_n. A monomial in R is a product x^ = x_1^ x_2^ \cdots x_n^ for an ''n''-tuple \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb^n of nonnegative integers. The following three conditions are equivalent for an ideal I \subseteq R: # I is generated by monomials, # If f = \sum_ c_ x^ \in I, then x^ \in I, provided that c_ is nonzero. # I is torus fixed, i.e, given (c_1, c_2, \dotsc, c_n) \in (\mathbb^*)^n, then I is fixed under the action f(x_i) = c_i x_i for all i. We say that I \subseteq \mathbb /math> is a monomial ideal if it satisfies any of these equivalent conditions. Given a monomial ideal I = (m_1, m_2, \dotsc, m_k), f \in \mathbb _1, x_2, \dotsc, x_n/math> is in I if and only if every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rees Algebra
In commutative algebra, the Rees algebra or Rees ring of an ideal ''I'' in a commutative ring ''R'' is defined to be R t\bigoplus_^ I^n t^\subseteq R The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined asR t,t^\bigoplus_^I^nt^\subseteq R ,t^This construction has special interest in algebraic geometry since the projective scheme defined by the Rees algebra of an ideal in a ring is the blowing-up of the spectrum of the ring along the subscheme defined by the ideal (see ).Eisenbud-Harris, ''The geometry of schemes''. Springer-Verlag, 197, 2000 Properties The Rees algebra is an algebra over \mathbb ^/math>, and it is defined so that, quotienting by t^=0 or ''t=λ'' for ''λ'' any invertible element in ''R'', we get \text_I R \ \leftarrow\ R t \to\ R. Thus it interpolates between ''R'' and its associated graded ring ''grIR''. * Assume ''R'' is Noetherian; then ''R t' is also Noetherian. The Krull dimension of the Rees algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formally Equidimensional
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] |
Multiplicity Of An Ideal
Multiplicity may refer to: Arts and entertainment * ''Multiplicity'' (film), a 1996 comedy film starring Michael Keaton * ''Multiplicity'' (album), 2005 studio album by Dave Weckl Science * Multiplicity (chemistry), multiplicity in quantum chemistry is a function of angular spin momentum * Multiplicity (informatics), a type of relationship in class diagrams for Unified Modeling Language used in software engineering * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (software), a software application which allows a user to control two or more computers from one mouse and keyboard * Multiplicity (statistical mechanics), the number of microstates corresponding to a particular macrostate in a thermodynamic system, symbolized by the Greek letter Ω * Dissociative identity disorder, psychological condition formerly called "multiple personality disorder" where a person exhibits multiple, distinct overlapping identities * Statistical mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind–Kummer Theorem
In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer. Statement for number fields Let K be a number field such that K = \Q(\alpha) for \alpha \in \mathcal O_K and let f be the minimal polynomial for \alpha over \Z /math>. For any prime p not dividing irreducible polynomials in \mathbb F_p /math>. Then (p) = p \mathcal O_K factors into prime ideals as (p) = \mathfrak p_1^ \cdots \mathfrak p_g^ such that N(\mathfrak p_i) = p^, where N is the ideal norm. Statement for Dedekind domains The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let \mathcal o be a Dedekind domain contained in its quotient field K, L/K a finite, separable field extension with L=Ktheta Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now known as Simons Laufer Mathematical Sciences Institute (SLMath). He served as Director of MSRI from 1997 to 2007, and then again from 2013 to 2022. Biography Eisenbud is the son of mathematical physicist Leonard Eisenbud, who was a student and collaborator of the renowned physicist Eugene Wigner. Eisenbud received his Ph.D. in 1970 from the University of Chicago, where he was a student of Saunders Mac Lane and, unofficially, James Christopher Robson. He then taught at Brandeis University from 1970 to 1997, during which time he had visiting positions at Harvard University, Institut des Hautes Études Scientifiques (IHÉS), University of Bonn, and Centre national de la recherche scientifique (CNRS). He joined the staff at MSRI in 1997, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
