Encyclopedia - Algebra Ring Theory: Encyclopedia Ii - Algebra Ring Theory - Formal Definition
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 multi...   » Read the article

Encyclopedia - Wolfgang Krull: Encyclopedia - Wolfgang Krull
Wolfgang Krull (1899 - 1971) was a German mathematician, working in the field of commutative algebra. See: Krull dimension, the Krull top...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia - Algebraic Geometry
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, wit...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia - Nicolas Bourbaki
Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books prese...   » Read the article

Encyclopedia - Closure Mathematics: Encyclopedia - Closure Mathematics
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 ...   » Read the article

Encyclopedia - Commutative Ring: Encyclopedia - Commutative Ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative la...   » Read the article

Encyclopedia - Coherent Sheaf: Encyclopedia - Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a...   » Read the article

Encyclopedia - Alexander Grothendieck: Encyclopedia - Alexander Grothendieck
Alexander Grothendieck (born March 28, 1928) was one of the most important mathematicians active in the 20th century. He was also one of ...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia - Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist an...   » Read the article

Encyclopedia - Algebra Ring Theory: Encyclopedia Ii - Algebra Ring Theory - Algebra Homomorphisms
An algebra homomorphism between two R-algebras is just an R-linear ring homomorphism. Explicity, is an algebra homomorphism if The...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Early Work
From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variable...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - 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...   » Read the article

Encyclopedia - Valuation Mathematics: Encyclopedia Ii - Valuation Mathematics - Model Theory
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 interpretatio...   » Read the article

Encyclopedia - Alexander Grothendieck: Encyclopedia Ii - Alexander Grothendieck - Mathematical Achievements
Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had been inven...   » Read the article

Encyclopedia - Even And Odd Functions: Encyclopedia Ii - Even And Odd Functions - Some Facts
Even and odd functions - Basic properties. The only function which is both even and odd is the constant function which is identically ...   » Read the article

Encyclopedia - Hilbert's Problems: Encyclopedia Ii - Hilbert's Problems - Nature And Influence Of The Problems
While there have been subsequent attempts to repeat the success of Hilbert's list, no other broadly-based set of problems or conjectures ...   » Read the article

Encyclopedia - Graded Algebra: Encyclopedia Ii - Graded Algebra - Graded Algebra
A graded algebra A is an algebra that has a direct sum decomposition such that Elements of An are known as homogeneous elements of ...   » Read the article

Encyclopedia - Local Ring: Encyclopedia Ii - Local Ring - Examples
Local ring - Commutative. All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings. To motivate...   » Read the article

Encyclopedia - Elementary Symmetric Polynomial: Encyclopedia Ii - Elementary Symmetric Polynomial - Definition
The elementary symmetric polynomials in n variables can be defined as and so forth, down to For each positive integer, at most n, ...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Affine Varieties
First we start with a field k. In classical algebraic geometry, this field was always , the complex numbers, but many of the same results...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Elementary Properties Of Polynomials
All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Motivation
In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such...   » Read the article

Encyclopedia - Scheme Mathematics: Encyclopedia Ii - Scheme Mathematics - History And Motivation
The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about ...   » Read the article

Encyclopedia - Gröbner Basis: Encyclopedia Ii - Gröbner Basis - Properties And Applications Of Gröbner Bases
Gröbner basis - Deciding equality of ideals. Reduced Gröbner bases can be shown to be unique for any given ideal and monomial orderin...   » Read the article

Encyclopedia - Mathematical Singularity: Encyclopedia Ii - Mathematical Singularity - Complex Analysis
In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a ...   » Read the article

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Topology And Smooth Structure
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a ...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Functions Between Two Manifolds
If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using loca...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Affine Varieties
First we start with a field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same result...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Representing Natural Numbers As Products Of Primes
The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique ...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Early Work
From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variable...   » Read the article

Encyclopedia - Radical Of An Ideal: Encyclopedia Ii - Radical Of An Ideal - Proof That The Radical Is An Ideal
Let a and b be in the radical of an ideal I. Then, for some positive integers m and n, an and bm are in I. We will show that a + b is in ...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - Books By Bourbaki
Aiming at a completely self-contained treatment of most of modern mathematics based on set theory, the group produced the following volum...   » Read the article

Encyclopedia - Differential Operator: Encyclopedia Ii - Differential Operator - Notations
The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include: ...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Elementary Properties Of Polynomials
All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - The Bourbaki Perspective, And Its Limitations
The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Gött...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Generalizations
The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Prime num...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Formulae Yielding Prime Numbers
Main article formula for primes There is no formula for primes which is more efficient at finding primes than the methods mentioned above...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Prime Gaps
Let pn denote the n-th prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the number of...   » Read the article

Encyclopedia - Scheme Mathematics: Encyclopedia Ii - Scheme Mathematics - Definitions
A scheme X is a locally ringed space with a covering by open sets Ui, such that the restriction of the structure sheaf OX to each Ui give...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Some Special Types Of Primes
A prime p is called primorial or prime-factorial if it has the form p = Π(n) ± 1 for some number n, where Π(n) stands for the product ...   » Read the article

Encyclopedia - Scheme Mathematics: Encyclopedia Ii - Scheme Mathematics - The Category Of Schemes
Schemes form a category if we take as morphisms the morphisms of locally ringed spaces. Morphisms from schemes to affine schemes are comp...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Functions Between Euclidean Spaces
Before giving a rigorous definition of a jet, it is useful to examine some special cases. Jet mathematics - Example: One-dimensional cas...   » Read the article

Encyclopedia - Radical Of An Ideal: Encyclopedia Ii - Radical Of An Ideal - Jacobson Radicals
Let R be any ring, not necessarily commutative. The Jacobson radical of R is the intersection of the annihilators of all simple right R-m...   » Read the article

Encyclopedia - Radical Of An Ideal: Encyclopedia Ii - Radical Of An Ideal - The Nilradical Of A Ring
Consider the set of all nilpotent elements of R, which will be called the nilradical of R (and will be denoted by N(R)). As can be easily...   » Read the article

Encyclopedia - Radical Of An Ideal: Encyclopedia Ii - Radical Of An Ideal - Definition
The radical of an ideal I in a commutative ring R, denoted by Rad(I) or √I, is defined as Intuitively, one can think of the radical ...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Foundational Work In Algebraic Geometry And The Weil Conjectures
In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundati...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Other Work
From 1959 onwards his interests turned towards number theory, in particular class field theory and the theory of complex multiplication. ...   » Read the article

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Vector Fields
A smooth assignment of a vector at each point of a manifold is called a vector field. Specifically, a vector field on a manifold M is a s...   » Read the article

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Examples
The simplest example is that of Rn. In this case the tangent bundle is trivial and isomorphic to R2n. Another simple example is the unit ...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Graphs
A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. T...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Sections
This subsection deals with the notion of jets of local sections a vector bundle. Almost everything in this section generalizes mutatis mu...   » Read the article

Encyclopedia - Graded Algebra: Encyclopedia Ii - Graded Algebra - G-graded Algebra
We can generalize the definition of a graded algebra to an arbitrary monoid G as an index set. A G-graded algebra A is an algebra with a ...   » Read the article

Encyclopedia - Gröbner Basis: Encyclopedia Ii - Gröbner Basis - Formal Definition
A Gröbner basis is characterised by any one of the following properties, stated relative to some monomial order: the ideal given by the...   » Read the article

Encyclopedia - Elementary Symmetric Polynomial: Encyclopedia Ii - Elementary Symmetric Polynomial - Properties
The elementary symmetric polynomials appear when we expand That is, when we substitute values for the variables , we obtain the univar...   » Read the article

Encyclopedia - Graded Algebra: Encyclopedia Ii - Graded Algebra - Graded Modules
The corresponding idea in module theory is that of a graded module, namely a module M over A such that also and This idea is much u...   » Read the article

Encyclopedia - Hilbert's Problems: Encyclopedia Ii - Hilbert's Problems - Summary
Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19 and 20 have a resolution that is accepted by consensus....   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets At A Point In Euclidean Space: Rigorous Definitions
This subsection focuses on two different rigorous definitions of the jet of a function at a point, followed by a discussion of Taylor's t...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Primality Tests
Main article primality test A primality test algorithm is an algorithm which tests a number for primality, i.e. whether the number is a p...   » Read the article

Encyclopedia - Scheme Mathematics: Encyclopedia Ii - Scheme Mathematics - Types Of Schemes
There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of sc...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Roots
A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial ...   » Read the article

Encyclopedia - Even And Odd Functions: Encyclopedia Ii - Even And Odd Functions - Even Functions
Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all real x: f(−x) = f(x) Ge...   » Read the article

Encyclopedia - Differential Operator: Encyclopedia Ii - Differential Operator - Coordinate-independent Description
In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential opera...   » Read the article

Encyclopedia - Differential Operator: Encyclopedia Ii - Differential Operator - Properties Of Differential Operators
Differentiation is linear, i.e., D(f + g) = (Df) + (Dg) D(af) = a(Df) where f and g are functions, and a is a constant. Any polynomia...   » Read the article

Encyclopedia - Differential Operator: Encyclopedia Ii - Differential Operator - Adjoint Of An Operator
Given a linear differential operator the adjoint of this operator is defined as the operator T * such that where the notation is u...   » Read the article

Encyclopedia - Even And Odd Functions: Encyclopedia Ii - Even And Odd Functions - Odd Functions
Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all real x: f(−x) = âˆ...   » Read the article

Encyclopedia - Hilbert's Problems: Encyclopedia Ii - Hilbert's Problems - The Problems As Hilbert's Manifesto
It is quite clear that the problem list, and its manner of discussion, were meant to be influential. Hilbert in no way fell short of the ...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Other Work
From 1959 onwards his interests turned towards number theory, in particular class field theory and the theory of complex multiplication. ...   » Read the article

Encyclopedia - Jean-pierre Serre: Encyclopedia Ii - Jean-pierre Serre - Foundational Work In Algebraic Geometry And The Weil Conjectures
In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundati...   » Read the article

Encyclopedia - Hilbert's Problems: Encyclopedia Ii - Hilbert's Problems - Summary
Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19 and 20 have a resolution that is accepted by consensus....   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - Dieudonné As Speaker For Bourbaki
Public discussion of, and justification for, Bourbaki's thoughts has in general been through Jean Dieudonné, who initially was the 'scri...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - The Bourbaki Perspective And Its Limitations
The underlying drive, in Weil and Chevalley at least, was the perceived need for French mathematics to absorb the best ideas of the Gött...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Regular Functions
Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manif...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Zeroes Of Simultaneous Polynomials
In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of al...   » Read the article

Encyclopedia - Alexander Grothendieck: Encyclopedia Ii - Alexander Grothendieck - Life
Alexander Grothendieck - Childhood and studies. Born to a Russian Jewish father and German Protestant mother in Berlin, he was a displa...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Elementary Properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero elem...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - The Category Of Affine Varieties
Using regular functions from an affine variety to , we can define regular functions from one affine variety to another. First we will def...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Projective Space
Consider the variety V(y=x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x,x2)...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - The Group
Accounts of the early days vary, but original documents have now come to light. The founding members were all connected to the Ecole Norm...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - Influence On Mathematics In General
The emphasis on rigour may be seen as a reaction to the work of Jules-Henri Poincaré, who stressed the importance of free-flowing mathem...   » Read the article

Encyclopedia - Algebraic Geometry: Encyclopedia Ii - Algebraic Geometry - Notes And History
Algebraic geometry was developed largely by the Italian geometers in the early part of the 20th century. Enriques classified algebraic su...   » Read the article

Encyclopedia - Local Ring: Encyclopedia Ii - Local Ring - Definition And First Consequences
A ring R is a local ring if it has one (and therefore all) of the following equivalent properties: R has a unique maximal left ideal. R ...   » Read the article

Encyclopedia - Local Ring: Encyclopedia Ii - Local Ring - Some Facts And Definitions
Local ring - Commutative. We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topologic...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Abstract Algebra
In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a fo...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Several Variables
In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can ...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Numerical Analysis
Polynomial - Polynomials and calculus. One important aspect of calculus is the project of analyzing complicated functions by means of a...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Examples
The function is an example of a cubic function with leading coefficient −7 and constant coefficient 3. ... See also:Polynomial, Po...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Divisibility
In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomial...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - More Variables
One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Open Questions
There are many open questions about prime numbers. The most significant of these is the Riemann hypothesis, which essentially says that t...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Finding Prime Numbers
The Sieve of Eratosthenes is a simple way and the Sieve of Atkin a fast way to compute the list of all prime numbers up to a given limit....   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - How Many Prime Numbers Are There?
There are infinitely many prime numbers. The oldest known proof for this statement is given by the Greek mathematician Euclid in his Elem...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Graphs
is a horizontal line with y-intercept a0. The graph of a degree 1 polynomial function (or linear function) f(x) = a0 + a1x, where is...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Polynomial Functions
For given constants (i.e., numbers) a0, â€¦, an in some field (possibly but not limited to R or C) with an non-zero, for n >...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Submodules And Homomorphisms
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Definition
Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplicati...   » Read the article

Encyclopedia - Mathematical Singularity: Encyclopedia Ii - Mathematical Singularity - Algebraic Geometry And Commutative Algebra
In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Types Of Modules
Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn in M such that every element of M is...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Relation To Representation Theory
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, su...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - More Advanced Examples Of Polynomials
Also in linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph the...   » Read the article

Encyclopedia - Module Mathematics: Encyclopedia Ii - Module Mathematics - Generalizations
Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covar...   » Read the article