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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] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Sheaf
In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf of finite rank and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry. For the theory of reflexive sheaves, one works over an integral noetherian scheme. A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive. Usually, the product of reflexive sheaves is defined as the reflexive hull of their tensor products (so the result is reflexive.) A coherent sheaf ''F'' is said to be "normal" in the sense of Barth if the restriction F(U) \to F(U - Y) is bijective for every open subset ''U'' and a closed subset ''Y'' of ''U'' of codimension at least 2. With this te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer. Examples The ring of entire functions on the open complex plane C form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring \mathbb /math> of integer polynomials is not . While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain . Many Prüfe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Ring
In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left coherent ring that is not left Noetherian. A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ... left modules is injective. References * * *{{eom, title=Coherent ring, first=V.E., last= Govorov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semihereditary Ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring ''R'' is called hereditary if all submodules of projective modules over ''R'' are again projective. If this is required only for finitely generated submodules, it is called semihereditary. For a noncommutative ring ''R'', the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring. To be left (semi-)hereditary, all (finitely generated) submodules of projective ''left'' ''R''-modules must be projective, and similarly to be right (semi-)hereditary all (finitely generated) submodules of projective ''right'' ''R''-modules must be projective. It is possible for a ring to be left (semi-)hereditary but not right (semi-)hereditary and vice versa. Equivalent definitions * The ring ''R'' is left (semi-)hereditary if and only if all ( finitely generated) left ideals of ''R'' are projective modules. * The ring ''R'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the same index set ''I'' such that ''B'' and ''B''∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ''V''∗. If it does span ''V''∗, then ''B''∗ is called the dual basis or reciprocal basis for the basis ''B''. Denoting the indexed vector sets as B = \_ and B^ = \_, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in ''V''∗ on a vector in the original space ''V'': : v^i\cdot v_j = \delta^i_j = \begin 1 & \text i = j\\ 0 & \text i \ne j\text \end where \delta^i_j is the Kronecker delta symbol. Introduction To perform operations with a vector, we must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion-free Module
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its zero element. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module. Examples of torsion-free modules Over a commutative ring ''R'' with total quotient ring ''K'', a module ''M'' is torsion-free if and only if Tor1(''K''/''R'',''M'') vanishes. Therefore flat modules, and in particular free and projective modules, are t ... [...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] |
