Let R be a commutative ring. An R-algebra is a set A which has the structure of both a ring and an R-module in such a way that ring multiplication is an R-bilinear map. Explicity, we must have If A itself is commutative (as a ring) then it is called a commutative R-algebra. Starting with an R-module A, we get an R-algebra by equipping A with an R-bilinear map A × A< ...

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Algebra ring theory, Algebra ring theory - Formal definition, Algebra ring theory - Algebra homomorphisms, Algebra ring theory - Examples, Algebra ring theory - Constructions

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Wolfgang Krull (1899 - 1971) was a German mathematician, working in the field of commutative algebra. See: Krull dimension, the Krull topology, Krull's intersection theorem and Krull's principal ideal theorem . ...

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Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find som ...

Including:

• Algebraic geometry - Zeroes of simultaneous polynomials
• Algebraic geometry - Affine varieties
• Algebraic geometry - Regular functions
• Algebraic geometry - The category of affine varieties
• Algebraic geometry - Projective space
• Algebraic geometry - The modern viewpoint
• Algebraic geometry - Notes and history

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Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. (Thus, an object is, among other things, a set.) An object is closed if it is equal to its closure. Typical structural properties of all closure operations are: The closure is increasing or extensive: the closure of an object contains the object. The closure is idempotent: the closure of the closure equals t ...

Including:

• Closure mathematics - Examples

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In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a. The study of commutative rings is called commutative algebra. Commutative ring - Examples. The most important example is the ring of integers with the two operations of addition and multiplication. Ordinary multiplication of inte ...

Including:

• Commutative ring - Examples
• Commutative ring - Constructing new commutative rings from given ones
• Commutative ring - General discussion

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In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

Including:

• Coherent sheaf - Coherent cohomology

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Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of its most extreme scientific personalities, with achievements over a short span of years that are still scarcely credible in their broad scope and sheer bulk, and an approach that antagonised even close followers. He made major contributions to algebraic geometry, homological algebra, and functional analysis. He was awarded the Fields Medal in 1966, and co-awarded the Crafoord Prize with Pierre Delig ...

Including:

• Alexander Grothendieck - Mathematical achievements
• Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles
• Alexander Grothendieck - Life
• Alexander Grothendieck - Childhood and studies
• Alexander Grothendieck - Politics and retreat from scientific community
• Alexander Grothendieck - Manuscripts written in the 1980s
• Alexander Grothendieck - Disappearance

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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Abelian category - Definitions. A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Pe ...

Including:

• Abelian category - Definitions
• Abelian category - Examples
• Abelian category - Elementary properties
• Abelian category - Related concepts
• Abelian category - History

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An algebra homomorphism between two R-algebras is just an R-linear ring homomorphism. Explicity, is an algebra homomorphism if The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg. The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing ...

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Algebra ring theory, Algebra ring theory - Formal definition, Algebra ring theory - Algebra homomorphisms, Algebra ring theory - Examples, Algebra ring theory - Constructions

Read more here: » Algebra ring theory: Encyclopedia II - Algebra ring theory - Algebra homomorphisms

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis refers to his dissertation on the Leray-Serre spectral sequence associated to a fibration. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently extravagant terms, and also made the point that the award ...

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Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - External link

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A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is ...

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Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

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In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. Informally, it is an assignment of particular values to the variables in a mathematical statement or equation. So for example the statement "x = y" is satisfied by (i.e. true for) valuations in which "x" is mapped to the same value as "y", and not satisfied by (i.e. false for) all other valuations. This may seem trivial in such a simple case, but is part of the process of for ...

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Valuation mathematics, Valuation mathematics - Model theory, Valuation mathematics - Algebra and algebraic geometry, Valuation mathematics - Examples

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Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been invented by Kiyoshi Oka and Jean Leray. Grothendieck took them to a higher level, changing the tools and the level of abstraction. Amongst his insights, he shifted attention from the study of individual varieties to the relative point of view (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. This he applied first to the Riemann-Roch theorem, aro ...

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Alexander Grothendieck, Alexander Grothendieck - Mathematical achievements, Alexander Grothendieck - Major mathematical topics from Récoltes et Semailles, Alexander Grothendieck - Life, Alexander Grothendieck - Childhood and studies, Alexander Grothendieck - Politics and retreat from scientific community, Alexander Grothendieck - Manuscripts written in the 1980s, Alexander Grothendieck - Disappearance

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Even and odd functions - Basic properties. The only function which is both even and odd is the constant function which is identically zero (i.e., f(x) = 0 for all x). In general, the sum of an even and odd function is neither even nor odd; e.g. x + x2. The sum of two even functions is even, and any constant multiple of an even function is even. The sum of two odd functions is odd, and any constant multiple of an odd function is ...

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Even and odd functions, Even and odd functions - Even functions, Even and odd functions - Odd functions, Even and odd functions - Some facts, Even and odd functions - Basic properties, Even and odd functions - Series, Even and odd functions - Algebraic structure

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While there have been subsequent attempts to repeat the success of Hilbert's list, no other broadly-based set of problems or conjectures has had a comparable effect on the development of the subject, or attained a fraction of its celebrity. For example the Weil conjectures are famous, but were rather casually announced. André Weil was perhaps temperamentally unlikely to put himself in the position of vying wi ...

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Hilbert's problems, Hilbert's problems - Nature and influence of the problems, Hilbert's problems - The problems as Hilbert's manifesto, Hilbert's problems - A round two dozen, Hilbert's problems - Summary, Hilbert's problems - Tabulated information, Hilbert's problems - Footnotes

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A graded algebra A is an algebra that has a direct sum decomposition such that Elements of An are known as homogeneous elements of degree n. An ideal, or other set in A, is homogeneous if for every element a it contains, the homogeneous parts of a are also contained in it. Since rings may be regarded as Z-algebras, a graded rin ...

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Graded algebra, Graded algebra - Graded algebra, Graded algebra - G-graded algebra, Graded algebra - Graded modules

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Local ring - Commutative. All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open interval around 0 of the real line. We are only interested in the local behavior of these functions near 0 and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an equivalence relation, an ...

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Local ring, Local ring - Definition and first consequences, Local ring - Examples, Local ring - Commutative, Local ring - Non-commutative, Local ring - Some facts and definitions, Local ring - Commutative, Local ring - General

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The elementary symmetric polynomials in n variables can be defined as and so forth, down to For each positive integer, at most n, there exists exactly one elementary symmetric polynomial of degree k in n variables. To form the one which has degree k, we form all products of k ...

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Elementary symmetric polynomial, Elementary symmetric polynomial - Definition, Elementary symmetric polynomial - Properties

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First we start with a field k. In classical algebraic geometry, this field was always , the complex numbers, but many of the same results are true if we assume only that k is algebraically closed. We define , called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, is, for the moment, just a collection of points. Henceforth we will drop the k in and instea ...

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Algebraic geometry, Algebraic geometry - Zeroes of simultaneous polynomials, Algebraic geometry - Affine varieties, Algebraic geometry - Regular functions, Algebraic geometry - The category of affine varieties, Algebraic geometry - Projective space, Algebraic geometry - The modern viewpoint, Algebraic geometry - Notes and history

Read more here: » Algebraic geometry: Encyclopedia II - Algebraic geometry - Affine varieties