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Topic: Indecomposable module

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	Indecomposable module - Encyclopedia, History, Geography and Biography
		In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.
		In many situations, all modules of interest can be written as direct sums of indecomposable ones; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied.
		By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent different from 0 and 1.
	www.arikah.com /encyclopedia/Indecomposable_module (430 words)

	Module (mathematics)
		A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R.
		An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules.
		An artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.
	www.xasa.com /wiki/en/wikipedia/m/mo/module__mathematics_.html (1181 words)

	Module (mathematics) (Site not responding. Last check: 2007-10-12)
		A free module is a module that has a or equivalently one that is isomorphic to direct sum of copies of the scalar ring R.
		An indecomposable module is a non-zero that cannot be written as a direct sum of two non-zero submodules.
		A faithful module M is one where the action of r in R gives an injective map M→M. Equivalently the annihilator of M is the zero ideal.
	www.freeglossary.com /Module_theory (1377 words)

	PlanetMath: indecomposable group (Site not responding. Last check: 2007-10-12)
		By definition, an indecomposable group is a nontrivial group that cannot be expressed as the internal direct product of two proper normal subgroups.
		An indecomposable module is a nonzero module that cannot be expressed as the direct sum of two nonzero submodules.
		This is version 5 of indecomposable group, born on 2005-07-16, modified 2005-12-23.
	www.planetmath.org /encyclopedia/IndecomposableModule.html (173 words)

	Fitting lemma - Encyclopedia, History, Geography and Biography
		If M is indecomposable and has finite length, then every endomorphism of M is either bijective or nilpotent.
		As an immediate consequence, we see that the endomorphism ring of every finite-length indecomposable module is local.
		Because M is indecomposable, one of those two summands must be equal to M, and the other must be equal to {0}.
	www.arikah.com /encyclopedia/Fitting_lemma (285 words)

	Injective module at AllExperts
		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.
	en.allexperts.com /e/i/in/injective_module.htm (1053 words)

	Springer Online Reference Works
		Every indecomposable representation of a finite-dimensional semi-simple algebra is equivalent to a direct summand of the regular representation.
		Basic problems studied in the representation theory of associative algebras are that of obtaining necessary and sufficient conditions for an algebra to belong to one of the types listed, as well as that of classifying the indecomposable representations in the finite and tame cases.
		It follows that the Grothendieck group of all finite-length modules modulo split exact sequences is the free Abelian group on the set of isomorphism classes of indecomposable representations.
	eom.springer.de /r/r081430.htm (1290 words)

	Springer Online Reference Works
		The endomorphism ring of a tilting module over a finite-dimensional hereditary algebra (cf.
		Riedtmann classification), is a tilting module, sometimes called APR-tilting module.
		The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre-injective direct summands simultaneously.
	eom.springer.de /t/t130100.htm (443 words)

	Simple module at AllExperts
		In abstract algebra, a (left or right) module S over a ring R is called simple or irreducible if it is not the zero module 0 and if its only submodules are 0 and S.
		The simple modules are precisely the modules of length 1; this is a reformulation of the definition.
		Every simple module is indecomposable, but the converse is in general not true.
	en.allexperts.com /e/s/si/simple_module.htm (481 words)

	Module (mathematics) - Definition, explanation
		The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f.
		It is a middle school mathematics module using the science and language of patterns to...
		It is a middle school mathematics module using the science and language of...
	www.calsky.com /lexikon/en/txt/m/mo/module__mathematics_.php (1308 words)

	Wolfgang Hassler's homepage-Research interests
		If one considers modules over rings that are slightly more complicated than Z, e.g., the ring Z[2i], where i denotes the imaginary unit, then it turns out that the structure of their finitely generated modules is tremendously complex.
		I am interested in the structure of finitely generated modules over commutative Noetherian rings, with a strong emphasis on the structure of indecomposable modules and non-uniqueness of direct-sum decompositions.
		For instance, the question which local rings admit finitely generated indecomposable modules having arbitrarily prescribed rank at the minimal primes was almost completely answered in the course of recent investigations.
	www.uni-graz.at /~hassler/research.htm (591 words)

	Homomorphisms
		A homomorphism from module M to module N is simply a matrix that commutes with the action of the algebra on M and N. Subsections
		The space of projective homomorphisms from module M to module N. That is, the space of all homomorphisms that factor through a projective module.
		Given an element x in a module over a basic algebra and a natural number n, the function returns the homomorphism from the n^(th) projective module for the algebra to the module with the property that the idempotent e of the projective module maps to x * e.
	www.umich.edu /~gpcc/scs/magma/text1008.htm (1383 words)

	Strongly Indecomposable (Site not responding. Last check: 2007-10-12)
		A left module m is strongly indecomposable if its endomorphisms, written on the right, form a local ring.
		The Z module Z, with endomorphism ring Z, is indecomposable, but not strongly indecomposable, since the ring of endomorphisms is not local.
		If m is a simple left r module then the ring of endomorphisms is a division ring.
	www.mathreference.com /ring-loc,strong.html (481 words)

	reptutorial
		This module of dimension 6 is in fact the Young module parametrized by the partition [2,2].
		We will conclude from the calculation that this permutation module is indecomposable with socle and top each the direct sum of 1 and 2, and that the socle equals the radical.
		We can only say that they are probably indecomposable because Decompose uses a probabilistic algorithm in which first a basis for the endomorphisms of the representation is computed and then 'random' endomorphisms are raised to a power larger than the dimension of the representation.
	www.math.umn.edu /~webb/GAPfiles/reptutorial.html (2022 words)

	[No title]
		This has led to a th* eory of varieties for infinitely generated modules* [3, 4], and the classification of thick subcat* egories of the stable category stmod(kG) of finitely generated modules, at least in the case of a p-g *roup [5].
		A module of the form T (* I) is determined by its Tate cohomology which is the coinduced module* ^I= Hom H(A;k)(H^(A; k) *; I).
		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)

	Indecomposable module - Slider (Site not responding. Last check: 2007-10-12)
		An abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form Z/p
		There are however abelian groups that cannot be written as a (finite or infinite) direct sum of indecomposables; the rational numbers Q form the simplest example.
		By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring contains an idempotent different from 0 and 1.
	enc.slider.com /Enc/Indecomposable (383 words)

	INDECOMPOSABLE MODULE (Site not responding. Last check: 2007-10-12)
		Every finitely generated abelian group is a direct sum of indecomposable abelian groups.
		There are however abelian groups that cannot be written as a direct sum of indecomposables; the rational numbers Q form the simplest example.
		In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique.
	www.yotor.org /wiki/en/in/Indecomposable%20module.htm (395 words)

	Lee Lady: Finite Rank Torsion Free Modules over Dedekind Domains (a book)
		In the realm of torsion free modules, however, the idea that results for abelian groups will automatically hold for modules over arbitrary principal ideal domains is simply not tenable.
		The transition from modules over principal ideal domains to modules over dedekind domains is actually a much smaller leap than that from the integers to an arbitrary PID.
		Unlike the theory of torsion groups, the theory of finite rank torsion free modules is becoming something that fits in fairly well with the mainstream of commutative ring theory.
	www.math.hawaii.edu /~lee/book (629 words)

	exchangepagina
		Since many modules met in practice have indecomposable decompositions, a natural problem arises: when such a decompositon is unique up to an equivalence, or, more generally, when two direct decompositions have equivalent refinements.
		We show that the hypotheses of Azumaya's Theorem can be replaced by those that the summands have the exchange property, we show that for indecomposable modules the 2-exchange property is equivalent to the finite exchange property, and we prove that a finite direct sum of modules with the exchange property has it also.
		In sections 4, 5, 6 we present the known classes of modules with the exchange property: strongly-invariant submodules of pure-injective modules, injective modules and their generalizations (quasi-injective and continuous modules), some classes of projective modules.
	www.mast.queensu.ca /~alina/exchangepagina/exchangepagina.html (362 words)

	Matches for: Author/Editor=(Happel_Dieter)
		In this book, the authors generalize with respect to a tilting module of projective dimension at most one for an artin algebra to tilting with respect to a torsion pair in an abelian category.
		A general theory is developed for such tilting and the reader is led to a generalization for tilted algebras which the authors call "quasitilted algebras".
		This class also contains the canonical algebras, and the authors show that the quasitilted algebras are characterized by having global dimension at most two and each indecomposable module having projective dimension at most one or injective dimension at most one.
	www.mathaware.org /bookstore?arg9=Dieter_Happel&fn=100&l=20&pg1=CN&r=1&s1=Happel%5FDieter (146 words)

	Amazon.com: "indecomposable representations": Key Phrase page (Site not responding. Last check: 2007-10-12)
		Lemma 1.1.1 A k-representation U of Sn is indecomposable if and only if O and I are the only idempotents...
		This approach also gave rise to a way of constructing all the indecomposable representations of the Dynkin diagrams by means of 'Coxeter functors' or 'reflection functors',...
		indecomposable representations 1 0 0 0 1 0 I I X F-, 0 1 0 y F-~ 0 0 1 0 I...
	www.amazon.com /phrase/indecomposable-representations (581 words)

	ABSTRACT ALGEBRA ON LINE: Modules (part 2)
		M is said to be indecomposable if its only direct summands are (0) and M. As our first example, we note that Z is indecomposable as a module over itself, since the intersection of any two nonzero ideals is again nonzero.
		Any submodule of a semisimple module has a complement that is a direct sum of minimal submodules.
		The first step is to show that any finitely generated torsion module can be written as a direct sum of finitely many indecomposable modules, and this is a consequence of the next propositions.
	www.math.niu.edu /~beachy/aaol/modules2.html (1490 words)

	MODULE INDéCOMPOSABLE (Site not responding. Last check: 2007-10-12)
		Dans beaucoup de situations, tous les modules d'intérêt peuvent être écrits en tant que sommes directes de les indécomposables ; les modules indécomposables peuvent alors être considérés comme "blocs fonctionnelle de base", les seuls objets qui doivent être étudiés.
		En regardant l'anneau d'endomorphism d'un module, on peut indiquer si le module est indécomposable : si et seulement si l'anneau d'endomorphism contient une quantité différente de 0 et de 1.
		Un module de longueur finie est indécomposable si et seulement si son anneau d'endomorphism est local.
	www.faktis.com /wiki/fr/mo/Module%20ind%E9composable.htm (458 words)

	Proceedings of the American Mathematical Society
		This is an analogue to a result in a complete module category proved by Happel and Liu.
		M.Auslander and I.Reiten, Modules determined by their composition factors, Illinois Journal of Mathematics, 29(2), 1985, 280-301.
		T.Wakamatsu, Stable equivalence of self-injective algebras and a generalization of tilting modules, J. Algebra, 134(1990), 298-325.
	www.ams.org /proc/2001-129-01/S0002-9939-00-05518-0/home.html (335 words)

	sesionL1 (Site not responding. Last check: 2007-10-12)
		If the given algebra is finitely generated then every differential left ideal is generated by constants, a non-commutative Tailor series decomposition formula is valid, and the category of locally nilpotent modules over the operator algebra is semisimple with the only simple object that is isomorphic to the optimal algebra as a module.
		We classify artin algebras that, together with their opposite algebras, have the property that every submodule of an indecomposable projective module is either projective or simple.
		Approximations with modules having linear resolutions I. The aim of the talk is to present some new results on modules over Koszul algebras.
	www.matmor.unam.mx /eventos/ams-smm/EN/SESIONES/sesionL1 (1530 words)

	Art of Problem Solving Forum (Site not responding. Last check: 2007-10-12)
		I did some inquiries and found that every simple module is indecomposable but not the converse.
		Then I was hoping to find a nonsimple indecomposable module, but that is no good either, as a module is indecomposable if and only if it only has zero and identity as idempotent endomorphisms.
		This in indecomposable but not irreducible as it has two factors that are both trivial.
	www.artofproblemsolving.com /Forum/topic-76629.html (474 words)

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