
	Where results make sense

Topic: Injective module

Related Topics

Pure injective module

Projective module

Direct summand

Injective function

Exact functor

Surjection

Open set

Set theoretic intersection

List of differential geometry topics

Troy, Tennessee

Tristan Honsinger

Marion, Wisconsin

Heads of Government of Equatorial Guinea

Law of conservation of matter

Baron Howard of Charlton

In the News (Tue 1 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Injective module - Wikipedia, the free encyclopedia
		In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
		Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q.
		These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functors.
	www.wikipedia.org /wiki/Injective_module (986 words)

	Injective module -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Injective modules were introduced by (Click link for more info and facts about Reinhold Baer) Reinhold Baer in 1940.
		In general, submodules, factor modules or infinite (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sums of injective modules need not be injective.
		These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as (Click link for more info and facts about derived functor) derived functors.
	www.absoluteastronomy.com /encyclopedia/i/in/injective_module.htm (1109 words)

	Algebraically compact module - Wikipedia, the free encyclopedia
		In abstract algebra, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows to solve infinite systems of equations in the module by finitary means.
		More generally, every injective module is algebraically compact, for the same reason.
		If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all group homomorphisms from H to Q/Z.
	en.wikipedia.org /wiki/Algebraically_compact_module (578 words)

	Opposite Algebras (Site not responding. Last check: 2007-11-07)
		Furthermore the dual of a projective OA-module is an injective A-module and the dual of a projective OA-resolution of a module M is an A-injective resolution of the dual of M. Subsections
		Given a module M defined over a basic algebra M, this function returns the dual of M as a module over the opposite of the algebra of M. Note that the opposite of the algebra of M is created if it does not already exist.
		Injective hulls, and injective resolutions of a module are computed by taking the projective cover or projective resolution of the dual module over the opposite algebra and then again taking the dual to retrieve modules or complexes over the original algebra.
	magma.maths.usyd.edu.au /magma/htmlhelp/text858.htm (747 words)

	Injective module (Site not responding. Last check: 2007-11-07)
		In mathematics, an injective module is a module Q that shares certain desirable properties withthe Z-module Q of all rationalnumbers.
		If X and Y are left-R modules and f : X → Y is an injective module homomorphism and g : X → Q is anarbitrary module homomorphism, then there exists a module homomorphism h : Y → Q such thathf = g, i.e.
		These injective resolutions are used to define the injective dimension of amodule (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functors.
	www.therfcc.org /injective-module-210887.html (919 words)

	Injective module (Site not responding. Last check: 2007-11-07)
		To show that a given module is injective, the following Injective Test Lemma is useful: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of R can be extended to all of R.
		It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left R-modules has enough injective." To prove this, one uses the peculiar properties of the abelian group Q/Z to construct an injective cogenerator in the category of left R-modules.
		One also talks about injective objects in categories more general then module categories, for instance in functor categories or in categories of sheaves of O
	www.sciencedaily.com /encyclopedia/injective_module (1018 words)

	[No title]
		The injective unstable modules are divided i* nto two classes: The reduced injective* modules and the nilpotent injective modules.* * A reduced injective module is a direct sum of direct factors of polynomial algebr* *as.
		Modules of type 0 and 1 are precisely the reduced modules* * and the nilclosed modules respectively.
		The module L(2) h* as a set of basis consisting of polynomials of the form uivj+ujvi for i 6= j and i *+j 4.
	hopf.math.purdue.edu /PetersonC-Jmoui/Inj_res_of_unst_mod.txt (4248 words)

	Projective module -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Using a basis of a free module F, it is easy to see that if we are given a surjective module homomorphism from N to M, the corresponding mapping from Hom(F,N) to Hom(F,M) is also surjective (it's from a product of copies of N to the product with the same index set for M).
		Submodules of projective modules need not be projective; a ring R for which every submodule of a projective left module is projective is called left hereditary.
		Every module over a (A piece of land cleared of trees and usually enclosed) field or (Click link for more info and facts about skew field) skew field is projective (even free).
	www.absoluteastronomy.com /encyclopedia/p/pr/projective_module.htm (831 words)

	Projective and Injective Modules (Site not responding. Last check: 2007-11-07)
		Modules, or unitary modules, can act as objects in a category, with r module homomorphisms acting as morphisms.
		A module p is projective if, for any pair of modules a and b, and any epimorphism f from a onto b, and any homomorphism g from p into b, there is at least one homomorphism h from p into a such that hf = g.
		A module j is injective if, for any pair of modules a and b, and any monomorphism f from a to b, and any homomorphism g from a to j, there is at least one homomorphism h from b to j such that fh = g.
	www.mathreference.com /mod-pit,intro.html (460 words)

	[No title]
		Pure injective modules have been studied for some time in representation theo* ry of finite di- mensional algebras, mostly because certain infinitely generated pure injectives * (so-called generic modules) control the representation type of an algebra [11].
		Injective modules over the cohomology ring Before introducing and investigating the functor T, we need to study the inj* ective modules* over the ordinary cohomology ring and also over the Tate cohomology ring.
		We d* enote by k the trivial A-module* and the cohomology ring H(A; k) is by definition ExtA(k; k).* * This is a finitely generated graded commutative k-algebra by a theorem of Friedlander and Suslin [* *13]; in particular, it is a Noetherian ring.
	hopf.math.purdue.edu /Benson-KrauseH/pureinj.txt (9989 words)

	Surjection Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		A function is injective (one-to-one) if each image is mapped to by at most one element of the domain.
		A note on terminology: a one-to-one function is injective, but may fail to be surjective, while a one-to-one correspondence is both injective and surjective.
		A function f : A → B is injective if and only if A is empty or f is left-invertible, that is, there is a function g: B → A such that g o f = identity function on A.
	www.karr.net /encyclopedia/Surjection (1169 words)

	Pure injective module (Site not responding. Last check: 2007-11-07)
		In abstract algebra, algebraically compactmodules, also called pure-injective modules, are modules that have a certain "nice" property which allows to solve infinite systems of equations in themodule by finitary means.
		A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C⊗ K is injective for every right R-module C.
		If H is a right module over the ring R, one forms the (algebraic)character module H* consisting of all grouphomomorphisms from H to Q/Z.
	www.therfcc.org /pure-injective-module-329438.html (535 words)

	Bijection, injection and surjection - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-11-07)
		A bijection is a function that is both injective and surjective.
		An injection, injective function or one-to-one function is a function which maps distinct arguments to distinct images.
		More formally a function f : A → B is injective if, for every x and y in its domain A, if x $\ne$ y then also f(x) $\ne$ f(y).
	www.arikah.net /encyclopedia/1-to-1 (770 words)

	Rings and Modules, MAS427
		Module theory was built up during the first half of the twentieth century, to collect together algebraic ideas that were important for various applications in group theory, number theory, geometry and algebraic topology among other places.
		But the theory of modules makes good sense on its own, and it is one of the most elegant parts of modern algebra.
		Introduction to module theory, starting from the definition of module: free, flat, projective and injective modules, products, coproducts, tensor products, exactness and the Hom functor will be covered.
	www.maths.qmw.ac.uk /~bill/MAS427.html (648 words)

	Novedad
		Let T be a hereditary torsion theory on the module category R-Mod and let A be a module.
		A module D is called A-T-divisible if for every T-closed (T-saturated) submodule B of A, every homomorphism B-->D extends to a homomorphism A-->D. The notions of T-divisible and self-T-divisible module are those naturally deduced from the above definition.
		We show that a finite direct sum of relatively injective modules is self-T-divisible if and only if each direct summand is self-T-divisible.
	www.um.es /matematicas/novedades/2004.02.13.crivei.html (105 words)

	categories: Functorial injective hulls (Site not responding. Last check: 2007-11-07)
		Apparently, therefore, the meaning of injective is a mutation obtained by changing the word "monic" in the above description to something stronger, such as "extremal monic" or "regular monic".
		In the days when all categories were abelian (that is, in the days when people actually talked about injective hulls) it was also the case that all monic-epics were isos, and this easy proof was a pretty standard exercise.
		The definition of injective hull forces E(B) --> B to be monic which, in turn, forces u_B to be an iso.
	north.ecc.edu /alsani/ct99-00(8-12)/msg00129.html (447 words)

	PlanetMath: injective module (Site not responding. Last check: 2007-11-07)
		is an injective module if it satisfies the following equivalent conditions:
		Cross-references: monomorphism, functor, short exact sequence, equivalent, satisfies, module
		This is version 4 of injective module, born on 2001-12-12, modified 2004-03-11.
	planetmath.org /encyclopedia/InjectiveModule.html (81 words)

	Módulo de Injective (Site not responding. Last check: 2007-11-07)
		De mostrar que un módulo dado es injective, el lema siguiente de la prueba de Injective es útil: un R izquierdo- el módulo Q es injective si y solamente si cualquier homomorfismo g: &rarr I; Q definido en un I ideal izquierdo de R se puede ampliar a todo el R.
		Es un cogenerator injective en la categoría de grupos abelian, que significa que es injective y cualquier otro módulo está contenido en un producto convenientemente grande de copias de Q/Z.
		Se utilizan estas resoluciones injective de definir la dimensión injective de un módulo (la longitud de la resolución injective más corta que termina en ceros, si existe una resolución tan finita) así como functors derivados.
	www.yotor.net /wiki/es/m%f3/M%F3dulo%20de%20Injective.htm (1002 words)

	Injective Module Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For injective module - Find injective module and more at Lycos Search.
		Find injective module - Your relevant result is a click away!
		Look for injective module - Find injective module at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Injective_module (1131 words)

	On Summand Sum and Summand Intersection
		In this work extending modules and lifting modules with the SSP (or SIP) are studied.
		Also we show that for an extending module M, M is UC-module if and only if M has the SIP if and only if M has the SSIP.
		-complemented modules, whose importance is due to their connection with extending modules.
	www.math.metu.edu.tr /seminars/antalya/absIII.shtml (5041 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Let M be a generalized Verma module, that is to say a U(g) module obtained by induction from a one dimensional p module.
		First it is shown that A(M) is injective, whenever M is a simple module.
		The main application of graded injectivity is to compute the multiplicity of a simple module in each filtration degree of F(M).
	www.math.jussieu.fr /gdralgebre/carquois2002/resumes/joseph.html (353 words)

	Cauchy modules (Site not responding. Last check: 2007-11-07)
		A module P is called projective when, for all surjective module morphisms
		A module P is projective iff P is a retract of some free module F.
		Show that a module P is finitely generated and projective if and only if P is a retract of a free module on a finite set.
	www-texdev.ics.mq.edu.au /Quantum/Sect5/Sect5.html (188 words)

	categories: injective modules in a topos (Site not responding. Last check: 2007-11-07)
		A week or so ago, I asked the question about injectives in a category of modules in over a ring object in a Grothendieck topos.
		I asked whether if I is an injective module and E is an object of the topos, I^E is injective.
		That is, if A and B are modules, then there is an object of the topos A -o B that is the subobject of B^A consisting of the additive morphisms.
	north.ecc.edu /alsani/ct01(9-12)/msg00002.html (162 words)

	Injective Hulls are not Natural - Ad'amek, Herrlich, Rosick'y, Tholen (ResearchIndex)
		Abstract: In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective.
		In particular, assigning to a eld its algebraic closure, to a poset or Boolean algebra its MacNeille completion, and to an R- module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.
		Ad'amek, H. Herrlich, J. Rosick'y, and W. Tholen, Injective hulls are not natural, preprint (Toronto, 1999).
	citeseer.ist.psu.edu /616202.html (573 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		There is a well-developed analysis in terms of the socle series, and an intimate connection with the localization theory of the ring.
		In first-order logic, the complete theory of an indecomposable injective (right) module over a (right) noetherian ring falls into one of the simplest and most well behaved model-theoretic classifications: they are "non-multidimensional totally transcendental".
		This series is an ascending sequence of definably closed submodules whose union exhausts the module, but we know nothing in general about its structure beyond that one simple fact.
	www.math.ucalgary.ca /~nikolaev/talks2003.html (1476 words)

	Math 421/621
		We showed in class that any projective module is the summand of a free module.
		Show the converse, that is, show that if P is the summand of a free module (i.e.
		there is a module K and a free module F such that
	www.ndsu.nodak.edu /ndsu/coykenda/M421-621.3.S2000.htm (74 words)

Try your search on: Qwika (all wikis)