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Ring Extension
In commutative algebra, a ring extension is a ring homomorphism R\to S of commutative rings, which makes an -algebra. In this article, a ring extension of a ring ''R'' by an abelian group ''I'' is a pair of a ring ''E'' and a surjective ring homomorphism \phi:E\to R such that ''I'' is isomorphic (as an abelian group) to the kernel of \phi. In other words, :0 \to I \to E \overset R \to 0 is a short exact sequence of abelian groups. (This makes ''I'' a two-sided ideal of ''E''.) Given a commutative ring ''A'', an ''A''-extension is defined in the same way by replacing "ring" with " algebra over ''A''" and "abelian groups" with "''A''-modules". An extension is said to be ''trivial'' if \phi splits; i.e., \phi admits a section that is an algebra homomorphism. This implies that ''E'' is isomorphic to the direct product of ''R'' and ''I''. A morphism between extensions of ''R'' by ''I'', over say ''A'', is an algebra homomorphism ''E'' → ''E'' that induces the identities on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, for example A \oplus B \oplus C, provided A, B, and C are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spa ... [...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 Director of the Mathematical Sciences Research Institute (MSRI); he previously served as Director of MSRI from 1997 to 2007. 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, and took a position at Berkeley at the same time. From 2003 to 2005 Eisenbud was President of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Group
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let ''R'' be a ring and let ''R''-Mod be the category of modules over ''R''. (One can take this to mean either left ''R''-modules or right ''R''-modules.) For a fixed ''R''-mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Five Lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other. Statements Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups. : file:5 lemma.svg The five lemma states that, if the rows are exact, ''m'' and ''p'' are isomorphisms, ''l'' is an epimorphism, and ''q'' is a monomorphism, then ''n'' is also an isomorphism. The two four-lemmas state: Proof The method of proof we shall use is commonly referred to as diagram chasing. We shall prove the five lemma by individually proving each of the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF(x) * F(x + y) = F(x) + F(y) * F(xy) = F(x) F(y) The first two conditions say that is a ''K''-linear map (or ''K''-module homomorphism if ''K'' is a commutative ring), and the last condition says that is a (non-unital) ring homomorphism. If admits an inverse homomorphism, or equivalently if it is bijective, is said to be an isomorphism between and . Unital algebra homomorphisms If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism F:A\rightarrow B is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section (category Theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g. Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f: X\to Y is a split epimorphism with split monomorphism g: Y\to X, then X is isomorphic to the direct sum of Y and the kernel of f. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work. Properties * A section that is also an epimorphism is an isomorphism. Dually a retraction that is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Ring
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
