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Topic: Injective cogenerator

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	NationMaster - Encyclopedia: Injective module (Site not responding. Last check: 2007-10-26)
		In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
		It is an injective cogenerator in the category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z.
		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.nationmaster.com /encyclopedia/Injective-module (1028 words)

	Injective cogenerator
		In category theory, the concept of an injective cogenerator is motivated by some major and important examples, such as Pontryagin duality[?].
		As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group.
		The cogenerator Q/Z is quite useful in the study of modules over general rings.
	www.ebroadcast.com.au /lookup/encyclopedia/in/Injective_cogenerator.html (552 words)

	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.
		It is an injective cogenerator in the category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z.
		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.
	en.wikipedia.org /wiki/Injective_module (986 words)

	NationMaster - Encyclopedia: Category of abelian groups (Site not responding. Last check: 2007-10-26)
		The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
		The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.
		An object in Ab is injective if and only if it is divisible; it is projective if and only if it is a free abelian group.
	www.nationmaster.com /encyclopedia/Category-of-abelian-groups (766 words)

	Injective cogenerator - Wikpedia (Site not responding. Last check: 2007-10-26)
		In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality.
		As an example of a cogenerator in the same category, we have Q/Z, the rationals modulo the integers, which is a divisible abelian group.
		The Tietze extension theorem can be used to show that an interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Injective_cogenerator (534 words)

	math lessons - Injective module
		If X and Y are left-R modules and f : X → Y is an injective module homomorphism and g : X → Q is an arbitrary module homomorphism, then there exists a module homomorphism h : Y → Q such that hf = g, i.e.
		In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: 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.
		The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.
	www.mathdaily.com /lessons/Injective_module (995 words)

	CARL FAITH:Professor Emeritus, Mathematics, Rutgers University
		Injective cogenerator rings and a theorem of Tachikawa, II, Proc.
		Injective quotient rings of commutative rings, Module Theory: Papers and problems from the special session, 1977, Lecture Notes in Math., 700 (1979), 151-203.
		Linearly compact injective modules and a theorem of Vamos, Publicacions Matemàtiques, Universitat Autònoma de Barcelona (UAB), 30, (1986), 127-148.
	www.phoenix-designs.com /carlfaith/pub.htm (1573 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 (601 words)

	HJM, Vol. 31, No. 3, 2005
		On the Minimal Injective Cogenerator over Almost Perfect Domains, pp.
		The minimal injective cogenerator E(R/P) and its endomorphism ring A are investigated.
		The purpose of this paper is to further the study of weakly injective and weakly tight modules a generalization of injective modules.
	www.math.uh.edu /~hjm/Vol31-3.html (2096 words)

	Injective cogenerator (Site not responding. Last check: 2007-10-26)
		A generator of a category with a zero object is some object G so that every non-zero object H has some non-zero morphism f:G -> H. cogenerator is an object C such that every other nonzero object H has some nonzero morphism f:H -> C. (Note the reversed order).
		Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism f:Sum(G) -> H is a surjection; and one can form direct products of C until the morphism f:H-> Prod(C) is one to one.
		Being a cogenerator says precisely that H* is 0 if and only if H is zero.
	www.theezine.net /i/injective-cogenerator.html (546 words)

	Injective module - free-definition (Site not responding. Last check: 2007-10-26)
		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.free-definition.com /Injective-object.html (977 words)

	Algebraically compact module (Site not responding. Last check: 2007-10-26)
		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.
		More generally, every injective module is algebraically compact, for the same reason.
		Algebraically compact modules share many other properties with injective objects because of the following: there exists exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.
	www.abitabouteverything.com /files/a/al/algebraically_compact_module.html (590 words)

	[No title]
		In A, being injective is equivalent to being a retract of such a functor.
		Moreover, the image of J is the subcategory 22 J. of injective objects in B, which is the same as the subcategory of projective ob- jects.
		As projectives are injective, the __ sequence splits, so X is a retract of P and thus is projective.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		The homotopy category of injectives We fix a locally noetherian Grothendieck category A. Thus A is an abelian Gro* then- dieck category and has a set A0 of noetherian objects which generate A, that is *, every object in A is a quotient of a coproduct of objects in A0.
		A coproduct of injective resolutions is again an injectiv* *e resolution, and the left adjoint I~ preserves all coproducts.
		Gorenstein injective approximations and Tate cohomology Let A be a locally noetherian Grothendieck category and suppose that the deri* *ved category D(A) is compactly generated.
	hopf.math.purdue.edu /KrauseH/stable.txt (12046 words)

	categories: injective modules in a topos (Site not responding. Last check: 2007-10-26)
		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.
		So what I am asking is whether for an injective I, the induced B -o I --> A -o I is epic.
	north.ecc.edu /alsani/ct01(9-12)/msg00002.html (162 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Moreover, the image of J is the subcategory 22 J. of injective objects in B, which is the same as the subcategory of projective o* *b- jects.
		Proposition 7.1.In B, any object of finite projective or injective dimension is both projective and injective, and thus lies in the image of J. Proof.Suppose that X has projective dimension at most n > 0.
		As projectives are injective,* * the_ sequence splits, so X is a retract of P and thus is projective.
	hopf.math.purdue.edu /Christensen-Strickland/phantoms.txt (10943 words)

	What is Category of abelian groups? : Abaara fun facts and uncommon knowledge (Site not responding. Last check: 2007-10-26)
		injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the
		The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint.
		The category has a projective generator (Z) and an injective cogenerator (Q/Z).
	www.abaara.com /pac/Category_of_abelian_groups (509 words)

	Cassels, J. W. S. - FPF Ring Theory Books at Real Groovy New Zealand
		A number of important classes of FPF rings are completely characterised including semiprime Neotherian, semiperfect Neotherian, perfect nonsingular prime, regular and self-injective rings.
		Finite group rings over PF or commutative injective rings are FPF.
		This work is the culmination of a decade of research and writing by the authors and includes all known theorems on the subject of noncommutative FPF rings.
	www.realgroovy.co.nz /books/isbn/0521277388 (539 words)

	Talk:Injective cogenerator - Wikpedia (Site not responding. Last check: 2007-10-26)
		User:Charles Matthews This rather isolated page is densely written, and really needs to be taken in hand.
		The example about cogenerators in a category of topological spaces doesn't quite fit the definition, as the category doesn't have a zero object.
		OK - this is mentioned in the book of Barr and Wells as an example, but I was worrying about whether it was quite right.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Talk:Injective_cogenerator (74 words)

	Representations of Algebras 2003
		Abstract: For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Morita-like equivalence, known as Foxby equivalence, between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension.
		The reason for introducing this notion is to extend the theory of Koszul duality for operads to the situation involving both operations and cooperations.
		We will discuss the general problem of determining when two artin algebras have isomorphic trivial extensions, which is of interest in tilting theory.
	mystic.math.neu.edu /alexmart/MADL/RT2003.html (1220 words)

	Abelian group
		Every field gives rise to two abelian groups in the same fashion.
		Another important example is the factor group Q/Z, an injective cogenerator.
		If n is a natural number and x is an element of an abelian group G, then nx can be defined as x + x +...
	www.teachersparadise.com /ency/en/wikipedia/a/ab/abelian_group.html (470 words)

	Proceedings of the American Mathematical Society (Site not responding. Last check: 2007-10-26)
		Abstract: We give new construction of injective resolutions of complexes and bimodules.
		Applying this construction to an injective resolution of a Noetherian ring, we construct a
		J. Xu, Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), 451-477.
	80-www.ams.org.library.uor.edu /proc/2000-128-08/S0002-9939-00-05305-3/home.html (268 words)

	CARL FAITH:Professor Emeritus, Mathematics, Rutgers University
		Matlis [58] proved that any injective module E over a Noetherian commutative ring R had this structure and that there is a 1-1 correspondence between prime ideals P and indecomposable injectives E(R/P).
		injective, then R C =R[[x]]) is a duo ring and Bezont.
		be a minimal injective resolution of the R-module M, and define the Noetherian depth of M, denoted n.d.M as the maximal i such that
	www.phoenix-designs.com /carlfaith/errata.htm (4113 words)

	Volume 39, number 1 (1995) (Site not responding. Last check: 2007-10-26)
		A ring $R$ is a right max ring if every right module $M\neq 0$ has at least one maximal submodule.
		It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module $E$ of $\operatorname{mod}$-$R$; also it suffices to check the submodules of the injective hull $E(V)$ of each simple module $V$ (Theorem 1).
		We characterize a right max ring $R$ via the endomorphism ring $\Lambda$ of any injective cogenerator $E$ of $\operatorname{mod}$-$R$; namely, $\Lambda/L$ has a minimal submodule for any left ideal $L=\operatorname{ann}_{\Lambda}M$ for a submodule (or subset) $M\ne 0$ of $E$ (Theorem 8.8).
	mat.uab.es /pubmat/v39(1)/39195_12.html (232 words)

	Amek - Expert Forums from Gig, EQ, Keyboard, Surround Professional
		Injective Hulls are not Natural - Ad'amek, Herrlich, Rosick'y
		In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are
		Taken straight from the AMEK 9098 console, this single-channel unit offers the The AMEK 9098CL retains all the character and musicality of the original
	waywide.com /wawd/amek.htm (292 words)

	Singular Manual: Control theory background
		The relationship between modules and behaviors is very rich and leads to deep results on system structure.
		The key to the module-behavior correspondence is a property of some signal spaces that are modules over the ring of differential (or difference) operators, namely, the injective cogenerator property.
		This property makes it possible to translate any statement on the solution spaces that can be expressed in terms of images and kernels, to an equivalent statement on the modules.
	www.mathematik.uni-kl.de /~zerz/CLIPS/sing_1123.htm (329 words)

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