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Dual Module
In mathematics, the dual module of a left (respectively right) module ''M'' over a ring ''R'' is the set of module homomorphisms from ''M'' to ''R'' with the pointwise right (respectively left) module structure. The dual module is typically denoted ''M''∗ or . If the base ring ''R'' is a field, then a dual module is a dual vector space. Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = \operatorname(A) is a finite commutative group scheme represented by a Hopf algebra Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedis ... ''A'' over a commutative ring ''k'', then the ... [...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] |
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] |
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] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Homomorphism
In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ''R''-''linear map'' if for any ''x'', ''y'' in ''M'' and ''r'' in ''R'', :f(x + y) = f(x) + f(y), :f(rx) = rf(x). In other words, ''f'' is a group homomorphism (for the underlying additive groups) that commutes with scalar multiplication. If ''M'', ''N'' are right ''R''-modules, then the second condition is replaced with :f(xr) = f(x)r. The preimage of the zero element under ''f'' is called the kernel of ''f''. The set of all module homomorphisms from ''M'' to ''N'' is denoted by \operatorname_R(M, N). It is an abelian group (under pointwise addition) but is not necessarily a module unless ''R'' is commutative. The composition of module homomorphisms is again a module homomorphism, and the identity map on a module is a module homomorphism. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. Pointwise operations Formal definition A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by Commonly, ''o'' and ''O'' are denoted by the same symbol. A similar definition is used for unary operations ''o'', and for operations of other arity. Examples \begin (f+g)(x) & = f(x)+g(x) & \text \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text \\ (\lambda \cdot f)(x) & = \lambda \cdot f(x) & \text \end where f, g : X \to R. See also pointwise product, and scalar. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Vector Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the ''continuous dual space''. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Homomorphism
In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of ''choices'' of canonical maps or canonical isomorphisms; for a typical example, see prestack. For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference. Examples *If ''N'' is a normal subgroup of a group ''G'', then there is a canonical surjective group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Module
Reflexive may refer to: In fiction: * Metafiction In grammar: *Reflexive pronoun, a pronoun with a reflexive relationship with its self-identical antecedent *Reflexive verb, where a semantic agent and patient are the same In mathematics and computer science: *Reflexive relation, a relation where elements of a set are self-related *Reflexive user interface, an interface that permits its own command verbs and sometimes underlying code to be edited *Reflexive operator algebra, an operator algebra that has enough invariant subspaces to characterize it *Reflexive space, a subset of Banach spaces *Reflexive bilinear form, a bilinear form for which the order of a pair of vectors does not affect whether it evaluates to zero. In biology *Reflexive antagonism, the phenomenon by which muscles with opposing functions tend to antagonistically inhibit each other. Other uses: * Reflexive Entertainment Reflexive Entertainment was a video game developer based in Lake Forest, California. The ... [...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] |
Torsionless Module
In abstract algebra, a module ''M'' over a ring ''R'' is called torsionless if it can be embedded into some direct product ''R''''I''. Equivalently, ''M'' is torsionless if each non-zero element of ''M'' has non-zero image under some ''R''-linear functional ''f'': : f\in M^=\operatorname_R(M,R),\quad f(m)\ne 0. This notion was introduced by Hyman Bass. Properties and examples A module is torsionless if and only if the canonical map into its double dual, : M\to M^=\operatorname_R(M^,R), \quad m\mapsto (f\mapsto f(m)), m\in M, f\in M^, is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive. * A unital free module is torsionless. More generally, a direct sum of torsionless modules is torsionless. * A free module is reflexive if it is finitely generated, but for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
