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Cosocle
In mathematics, the term cosocle (socle meaning ''pedestal'' in French) has several related meanings. In group theory, a cosocle of a group ''G'', denoted by Cosoc(''G''), is the intersection of all maximal normal subgroups of ''G''. Adolfo Ballester-Bolinches, Luis M. Ezquerro, ''Classes of Finite Groups'', 2006, ,p. 97/ref> If ''G'' is a quasisimple group, then Cosoc(''G'') = Z(''G''). In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.) In the context of module theory, the cosocle of a module over a ring ''R'' is defined to be the maximal semisimple quotient of the module. See also * Socle *Radical of a module In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socle (mathematics)
In mathematics, the term socle has several related meanings. Socle of a group In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator ''u'', which has two minimal normal subgroups, one generated by ''u''4 (which gives a normal subgroup with 3 elements) and the other by ''u''6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by ''u''4 and ''u''6, which is just the group generated by ''u''2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socle (mathematics)
In mathematics, the term socle has several related meanings. Socle of a group In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator ''u'', which has two minimal normal subgroups, one generated by ''u''4 (which gives a normal subgroup with 3 elements) and the other by ''u''6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by ''u''4 and ''u''6, which is just the group generated by ''u''2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Module
In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(''M'') of ''M''. Definition Let ''R'' be a ring and ''M'' a left ''R''-module. A submodule ''N'' of ''M'' is called maximal or cosimple if the quotient ''M''/''N'' is a simple module. The radical of the module ''M'' is the intersection of all maximal submodules of ''M'', :\mathrm(M) = \bigcap_ N Equivalently, :\mathrm(M) = \sum_ S These definitions have direct dual analogues for soc(''M''). Properties * In addition to the fact rad(''M'') is the sum of superfluous submodules, in a Noetherian module rad(''M'') itself is a superfluous submodule. * A ring for which rad(''M'') = for every right ''R''-module ''M'' is called a right V-ring. * For any module ''M'', rad(''M''/rad(''M'')) is zero. * ''M'' is a finitely generate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup). Given a module over a ring , and a submodule of , the quotient space is defined by the equivalence relation : a \sim b if and only if b - a \in B, for any in . The elements of are the equivalence classes = a+B = \. The function \pi: A \to A/B sending in to its equivalence class is called the ''quotient map'' or the ''projection map'', and is a module homomorphism. The addition operation on is defined for two equivalence clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings. For a group-theory analog of the same notion, see ''Semisimple representation''. Definition A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module ''M'', the following are equivalent: # ''M'' is semisimple; i.e., a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Theory
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 conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Postnikov
Mikhail Mikhailovich Postnikov (russian: Михаи́л Миха́йлович По́стников; 27 October 1927 – 27 May 2004) was a Soviet mathematician, known for his work in algebraic and differential topology. Biography He was born in Shatura, near Moscow. He received his Ph.D. from Moscow State University under the direction of Lev Pontryagin, and then became a professor at this university. He died in Moscow. Selected publications * ''Fundamentals of Galois theory'', 1961; 142 ppDover reprint, 2004* ''The variational theory of geodesics'', Translated from the Russian by Scripta Technica, Inc. Edited by Bernard R. Gelbaum. Saunders, Philadelphia, Pa., 1967; 200 pp. * ''Linear algebra and differential geometry''. Translated from the Russian by Vladimir Shokurov. Mir Publishers, 1982; 319 pp. * ''Smooth manifolds'', Mir Publishers, 1989; 511 pp. ''Geometry VI: Riemannian Geometry'' Springer, 2001, 504 pp. French * ''Leçons de géométrie : Semestre I : Géomét ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by conside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of "mathematical ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
