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Topic: Injective hull

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Injective resolution

Divisible module

	PlanetMath: injective hull
		is both an injective module and an essential extension of
		has an injective hull, which is unique up to isomorphism.
		This is version 3 of injective hull, born on 2002-01-05, modified 2004-06-07.
	planetmath.org /encyclopedia/InjectiveHull.html (81 words)

	Injective module (Site not responding. Last check: 2007-09-18)
		In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
		Injective modules were introduced by Reinhold Baer in 1940.
		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.worldhistory.com /wiki/I/Injective-module.htm (1047 words)

	Opposite Algebras (Site not responding. Last check: 2007-09-18)
		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
		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.
		The complex giving the minimal injective resolution of the module M together with the inclusion homomorphism from M into its injective hull.
	magma.maths.usyd.edu.au /magma/htmlhelp/text858.htm (747 words)

	categories: Functorial injective hulls
		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)

	[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].
		Given a homogeneous prime ideal p, the injective hull Ip = E(* R=p) of the quotient R=p is indecomposable, and each injective* indecomposable is isomorphic* * to a shifted copy Ip[n] for some prime p and some n 2 Z. A theorem of Matlis [23] describes the e* *ndomorphism ring of Ip[n].
		The degree n part of Ip[n] is equal to the injective hull of the quotient of th* *is local ring by its maximal ideal.
	hopf.math.purdue.edu /Benson-KrauseH/pureinj.txt (9989 words)

	categories: Re: Functorial injective hull.
		The conditions a) and b) are obviously satisfied in the case when E is "injective hull functor" (of coarse a) is true up to iso, again see P,Freyd proof).
		Hence by d) the injective dimension of C is 0 and we have the result again.
		Another words the injective dimension is the obstruction for nonnaturality of the injective hull.
	north.ecc.edu /alsani/ct99-00(8-12)/msg00135.html (569 words)

	[No title]
		Li* n [Lin76] showed that ssY is not an injective ssS-module if Y is not torsion, but did n *ot realize that completion would solve this problem if the generating hypothesis is true.
		Suppose E is a spectrum such that ssE is an injective* ss*S-module.
		To see that ssYp is injective* over ssSp, we use Baer's criterion, wh ich tells us we need only check that, given an ideal a of ssSp and a map f :a -!ss* *Yp, there is an extension ssSp -!ssYp of ssSp-modules.
	hopf.math.purdue.edu /Hovey/freyd.txt (5265 words)

	Injective hull - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-09-18)
		In mathematics, a module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective.
		For example, the injective hull of an integral domain is its field of fractions.
		This page was last modified 08:42, 15 September 2004.
	en.wikipedia.org /wiki/Injective_hull (66 words)

	Injective hull - TheBestLinks.com - Integral domain, Mathematics, Field of fractions, Module (mathematics), ... (Site not responding. Last check: 2007-09-18)
		Injective hull - TheBestLinks.com - Integral domain, Mathematics, Field of fractions, Module (mathematics),...
		Injective hull, Integral domain, Mathematics, Field of fractions, Module...
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	www.thebestlinks.com /Injective_hull.html (108 words)

	DOCUMENTA MATHEMATICA, Quadratic Forms LSU (2001), 121-139 (Site not responding. Last check: 2007-09-18)
		In contrast, the first paper that studied metric spaces {\em as such} -- without trying to study their embeddability into any one of the standard metric spaces nor looking at them as mere `presentations' of the underlying topological space -- was, to our knowledge, written in the late sixties by John Isbell.
		In particular, Isbell showed that in the category whose objects are metric spaces and whose morphisms are {\em non-expansive} maps, a unique {\em injective hull} exists for every object, he provided an explicit construction of this hull, and he noted that, at least for finite spaces, it comes endowed with an intrinsic polytopal cell structure.
		In this paper, we discuss Isbell's construction, we summarize the history of --- and some basic questions studied in --- {\it phylogenetic analysis}, and we explain why and how these two topics are related to each other.
	www.ii.uj.edu.pl /EMIS/journals/DMJDMV/lsu/dress-huber-multon.html (218 words)

	HJM, Vol. 30, No. 3, 2004
		The purpose of this paper is to further the study of weakly injective and weakly tight modules a generalization of injective modules.
		A right R-module M is said to be weakly tight (tight) if every finitely generated submodule N of its injective hull E(M) is embeddable in a direct sum of copies of M (is embeddable in M).
		For some classes K of modules, we study when direct sums of modules from K are weakly tight, tight, weakly injective.
	www.math.uh.edu /~hjm/Vol30-3.html (1888 words)

	Injective Modules Over Non-Artinian Serial Rings (Site not responding. Last check: 2007-09-18)
		Suppose the injective hull of each simple left R-module is uniserial.
		Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has idemposable injective left modules uniserial.
		The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.
	anziamj.austms.org.au /JAMSA/V44/Part2/Hill.html (154 words)

	Atlas: Injective hulls are not natural by Jiri Adamek (Site not responding. Last check: 2007-09-18)
		The formation of injective hulls in a category K (in which every object has an injective hull) is seldom natural in the sense of defining an endofunctor F of K together with a natutral transformation Id -> F. Thus neither MacNeille completion of posets, nor injective hull of modules are natural.
		There is, nevertheless, a general procedure of assigning naturally injective extensions to objects by using an iterative construction of "approximations" of injectivity.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-07.
	atlas-conferences.com /c/a/e/q/07.htm (155 words)

	Ladislav Bican (Site not responding. Last check: 2007-09-18)
		Abstract:Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers.
		In this note we shall study some properties of the lattice $\Cal E_{\tau }(M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective.
		The results obtained are applied to the study of relatively exact covers of torsionfree modules.
	www.math.ethz.ch /EMIS/journals/CMUC/cmuc0304/abs/bicanl.htm (113 words)

	Volume 39, number 1 (1995) (Site not responding. Last check: 2007-09-18)
		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)

	AMCA: Tight and cotight modules by Mohammad Saleh (Site not responding. Last check: 2007-09-18)
		\sigma[M] is said to be weakly injective (resp., weakly tight) in \sigma[M] if for every finitely generated submodule N of the M-injective hull [^Q], N is contained in a submodule Y of [^Q] such that Y =~ Q (resp., N is finitely Q-cogenerated).
		Several characterizations of semisimple modules are obtained using weak injectivity and weak projectivity in \sigma[M].
		In particular, we get necessary and sufficient conditions for \sum-tightness or \sum-weak tightness of the injective hull of a simple module.
	at.yorku.ca /c/a/b/w/59.htm (214 words)

	Ulrich Heckmanns (Site not responding. Last check: 2007-09-18)
		Ultrametric spaces are defined similar to non-Archimedean metric spaces, except that the set of values is an arbitrary partially ordered set (with a least element).
		With a suitable notion of morphisms we get a reasonable characterization of injective ultrametric spaces (with respect to embeddings), which is similar to the one in [1].
		Moreover, every ultrametric space has a uniquely determined injective hull (by a slight modification of this term).
	www.utm.edu /staff/jschomme/topology/c/a/a/e/33.htm (121 words)

	Weakly-Injective Modules Over Hereditary Noetherian Prime Rings (Site not responding. Last check: 2007-09-18)
		A module M is said to be weakly-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N
		In particular we show that torsion-free modules over bounded hnp rings are always weakly-injective, while torsion modules finite Goldie dimension are weakly-injective only if they are injective.
		Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective mudule has such a decomposition.
	anziamj.austms.org.au /JAMSA/V58/Part3/Jain.html (133 words)

	Atlas: Extending Modules Satisfying (S) by Ayse Cigdem Ozcan (Site not responding. Last check: 2007-09-18)*
		Leonard [L] called a (right) R-module M a small module if M a small submodule of its injective hull (i.e.
		Theorem 1 The following are equivalent for a ring R. i) Every injective R-module is lifting (a (right) H-ring).
		iii) R is right perfect and the injective hull of every semisimple module satisfies (S
	atlas-conferences.com /c/a/f/e/43.htm (328 words)

	YAYIN LÝSTESÝ (Site not responding. Last check: 2007-09-18)
		Modules with zero radical of their injective hull, Hacettepe Bulletin of Natural Sciences and Engineering, 27, 45-49 (1998).
		Modules with small cylic submodules in their injective hull, Comm.
		Direct sum of modules having (S*), East West J. Math, to appear
	www.mat.hacettepe.edu.tr /people/COzcan/makale/Cigdem.htm (124 words)

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