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Topic: Finitely generated module

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	Finitely-generated module - Wikipedia, the free encyclopedia
		In the case where the module M is a vector space over a field R, and the generating set is linearly independent, n is well-defined and is referred to as the dimension of M (well-defined means that any linearly independent generating set has n elements: this is the dimension theorem for vector spaces).
		Finitely generated modules over the ring of integers Z coincide with the finitely generated abelian groups; these are completely classified.
		However, for a general ring R, it is an important concept, as is evidenced by the fact that the category of all finitely presented R-modules is abelian, something that's not generally true for the category of all finitely-generated R-modules.
	en.wikipedia.org /wiki/Finitely-generated_module (778 words)

	Finitely-generated module (Site not responding. Last check: 2007-10-08)
		Thus we say that the whole area of the map is generated by the set together with coefficients from the real numbers.
		The set is referred to as a generating set for M in this case.
		R itself is a finitely-generated R-module [with as generating set].
	www.worldhistory.com /wiki/F/Finitely-generated-module.htm (817 words)

	PlanetMath: finitely generated module (Site not responding. Last check: 2007-10-08)
		is said to be finitely generated if there is a finite subset
		"finitely generated module" is owned by Thomas Heye.
		This is version 8 of finitely generated module, born on 2003-10-15, modified 2004-11-04.
	planetmath.org /encyclopedia/FinitelyGeneratedRModule.html (105 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Corollary.} {Over a PID a finitely generated module $_RM$ is torsion free iff it is free.} {\bf Proof.} We may assume that $M\not= 0$ is torsion free.
		Since the rank of a finitely generated free module completely determines the module, Theorem 5 shows that a complete analysis of finitely generated modules depends on a description of the torsion ones.
		Moreover, since $\Soc M$ is a submodule of the finitely generated module $M$, it is finitely generated by Corollary 2.
	darkwing.uoregon.edu /~anderson/math648/lecture12.html (1662 words)

	MC249 Rings and modules
		This leads firstly to the idea of a Euclidean domain, which gives a general setting for the division algorithms already seen in MC145 for the integers and polynomials over a field, and secondly to the concept of a principal ideal domain, which gives particularly useful structure theorems for both rings and modules.
		Describing the structure of rings and modules is important in abstract algebra and enables us to apply this theory to many diverse areas of mathematics; from the examples listed above, it can be seen that the ideas in this course are used in number theory, linear algebra and group theory.
		The aim of this course is to introduce students to the basic structure and theory of rings and modules and to develop this theory to investigate important classes of integral domains and the classification of any finitely generated module as a homomorphic image of a free module.
	www.mcs.le.ac.uk /Modules/Year3/MC249.html (611 words)

	Finitely generated abelian groups (Site not responding. Last check: 2007-10-08)
		We will now prove the structure theorem for finitely generated abelian groups, since it will be crucial for much of what we will do later.
		We first remark that any subgroup of a finitely generated free abelian group is finitely generated.
		Then we see that finitely generated abelian groups can be presented as quotients of finite rank free abelian groups, and such a presentation can be reinterpreted in terms of matrices over the integers.
	modular.fas.harvard.edu /papers/ant/html/node9.html (725 words)

	Swan's theorem - Wikipedia, the free encyclopedia
		Swan's theorem relates vector bundles to projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".
		Swan's theorem states that this module is finitely generated and projective over C
		Suppose X is a compact Hausdorff space, and C(X) is the ring of continuous real-valued functions on X.
	en.wikipedia.org /wiki/Swan's_theorem (193 words)

	[No title]
		Then the ideal of essential * classes in the mod-p cohomology ring of G is a Cohen-Macaulay module* whose Krull dimension is the p-rank of the centre of G. This basically answers in t* *he affirmative a question posed by J. Carlson (Question 5.4 in [7]).
		As H(C) is polynomial for p = 2 and polynomial tensor exterior for p odd, and the dimension is equal *to the rank of C in both cases, d is the p-rank of the centre of G. Remark.
		Recalling that H(C) is a polynomial algebra for p = 2, and polynomial tensor exterior for p odd, one ded uces that H(C) is a free and finitely generated R-module.
	hopf.math.purdue.edu /DJGreen/essCM.txt (2858 words)

	Finitely-generated module
		Take the rational numbers whose denominator is 1, 2, 3 or 6 (after simplification: 16/12=4/3, so it belongs to this set).
		A set of generators is, for example, {1/1,1/2,1/3,1/6}, but also {1/6} -which is, obviously, minimal-.
		Since it has n elements, we say that M is an n-generator module.
	www.sciencedaily.com /encyclopedia/finitely_generated_module (603 words)

	[No title]
		B '42 -4:3 ~14 has finite rank n, that n is the minimal dimension for a realization of the sequence, and that a system realizing the sequence has minimal dimension iff it is canonical, i.e.
		Let us first establish the general fact, of interest by itself, that any finitely generated torsion-free module X may be the state module of a canonical svstern.
		Ever_v finitely, generated refleviTe R-module is projective ~ff the global d7mension of R is inferior or equal to 2.
	www.math.rutgers.edu /~sontag/FTP_DIR/noether-realiz.txt (3820 words)

	Citations: Reductions of modules - Rees (ResearchIndex)
		For basic facts and terminology we shall use [3] and [5] As for notation: let (R, m) be a Noetherian local ring and let M be a finitely generated R module; we set #(M)for the minimal number of generators of M, #(M) for its length if M has a finite composition series.
		The Buchsbaum-Rim Polynomial of a Module - Brennan, Ulrich (2001)
		for a finitely generated module E such that K Omega R E is K free and the R map E K Omega R E is injective) as in this situation the.
	citeseer.ist.psu.edu /context/1268862/0 (879 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A projective module M that is finitely generated must be finitely presented and flat, and a finitely presented flat module is projective, so the problem is equivalent to finding a finitely generated flat module that is not finitely presented.
		In this ring, it is easy to see that any ideal of finite type is generated by an idempotent.
		A is not noetherian, so there exists an ideal J that is not of finite type.
	www.math.niu.edu /~rusin/known-math/01_incoming/flat_module (259 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Then K is a finitely generated free module which can be generated by n elements or less.
		Remark: it is obvious that a quotient module of a finitely generated module is finitely generated.
		Since a free module is torsion-free, the intersection of F and Mtor is 0.
	www.math.ucla.edu /~blasius/110bh.1.01w/modulesoverpids.doc (1101 words)

	PlanetMath: finitely generated projective module (Site not responding. Last check: 2007-10-08)
		"finitely generated projective module" is owned by mhale.
		Cross-references: inner product, projection, idempotent, finitely generated, ring, unital
		This is version 3 of finitely generated projective module, born on 2003-02-26, modified 2003-11-18.
	planetmath.org /encyclopedia/FinitelyGeneratedProjectiveModule.html (56 words)

	[No title]
		Suppose that W is a finite group of K-automorphisms of H* Bo and that R is the ring of invariants H* (Bo)W.
		This quotient ring is generated as an Decomposition Theorem 9 algebra by exterior generators {yi} of dimension 1 together with their Bockstein images {fiyi} in dimension 2.
		The subring S of Fp R HV generated by y and fiy is clearly closed under the action of Ap and isomorphic as an Ap algebra to HZ=p.
	hopf.math.purdue.edu /Dwyer-Wilkerson/cohomology-decompositions/jm.txt (6117 words)

	Indecomposable module (Site not responding. Last check: 2007-10-08)
		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.
		A module of finite length is indecomposable if and only if its endomorphism ring is local.
	www.worldhistory.com /wiki/I/Indecomposable-module.htm (456 words)

	(Site not responding. Last check: 2007-10-08)
		Let $M$ be a finitely generated module of finte projective dimension over a local ring $A$, and let $N$ be a finitely generated $A$-module such that $M\otimes N$ is a module of finite length.
		This is a result about intersections, as the hypothesis ``$M\otimes N$ is a module of finite length'' is a statement about the intersections of the supports of the modules $M$ and $N$.
		(Auslander) If $M$ is a module of finite projective dimension over a local ring $A$, and if an element $x$ in $A$ is not a zero-divisor on $M$, then $x$ is not a zero-divisor on $A$.
	www.math.uiuc.edu /~ssather/TEACH/syllabus_fa02.html (531 words)

	Univ at Albany: W. F. Hammond: Math 520B (Site not responding. Last check: 2007-10-08)
		Show that if P is a finitely generated projective R-module, then P^{} = Hom_{R}(P, R) is also a finitely* generated projective R-module.
		Let M and N be finitely generated R-modules for which there is an isomorphism M \otimes_{R} N ~= R.
		P is a finitely generated projective left R-module.
	math.albany.edu /~hammond/course/mat520bf99/assgt/a991206.html (184 words)

	Finitely-generated module - Encyclopedia Glossary Meaning Explanation Finitely-generated module (Site not responding. Last check: 2007-10-08)
		Finitely-generated module - Encyclopedia Glossary Meaning Explanation Finitely-generated module.
		Here you will find more informations about Finitely-generated module.
		The orginal Finitely-generated module article can be editet
	www.encyclopedia-glossary.com /en/Finitely-generated-module.html (819 words)

	Algebra Prelim, January 2005 (Site not responding. Last check: 2007-10-08)
		Let R be a commutative Noetherian ring with a 1 and let M be a finitely generated R-module.
		Suppose R is a principal ideal domain that is not a field, and that M is a finitely generated R-module.
		Let G be a finite group with a composition series of length 2.
	www.math.vt.edu /people/linnell/Teaching/Algprelims/Jan05 (178 words)

	The Structure of a Finitely Generated Module
		PID Modules, The Structure of a Finitely Generated Module
		In this case a cyclic module is the integers, or the integers mod n for some positive integer n.
		To summarize, a finitely generated module over a pid is the direct product of a free module whose rank is bounded by the number of generators, times a finite number of cyclic modules, which are quotients of various prime powers in r.
	www.mathreference.com /mod-pid,struct.html (733 words)

	Vector bundles as modules over the algebra of smooth functions
		What's a finitely generated projective module, and why should I care?" The point is that being a finitely generated projective module is the algebraic equivalent of the condition that a vector bundle over M looks locally like M x R^n.
		If we have an algebra A, a finitely generated projective module M is one for which there is some other module N with M + N isomorphic to A^n.
		In math lingo, a finitely generated projective module is just a "summand of a finitely generated free module" - that is, it appears as part of a direct sum decomposition of some module A^n.
	www.usenet.com /newsgroups/sci.physics.research/msg02836.html (676 words)

	Finitely Generated = Integral
		Using terminology from the world of modules, the annihilator of m is 0.
		Since m is an r module, multiplication by x in r multiplies the coefficients on the basis elements by x.
		Cross the generators of h over g with the generators of g over r to show h is a finitely generated r module.
	www.mathreference.com /id-ext,fingen.html (1342 words)

	Singular Manual - module (Site not responding. Last check: 2007-10-08)
		If M is a submodule of R^n, R the basering, generated by vectors v_1,..., v_k, then v_1,..., v_k may be considered as the generators of relations of R^n/M between the canonical generators
		Hence any finitely generated R-module can be represented in SINGULAR by its module of relations.
		module M=v1,...,vk; matrix A=M; creates the presentation matrix of size n x k for R^n/M, i.e., the columns of A are the vectors v_1,..., v_k which generate M (cf.
	www.dmi.units.it /assistenza/Singular/html/singular_96.html (113 words)

	Commutative Algebra Seminar, University of Nebraska-Lincoln
		Flat dimensions of non-finitely generated modules Abstract: Complete intersection dimension, respectively, Cohen-Macaulay dimension, of finitely generated modules were introduced by Avramov, Gasharov, and Peeva, respectively, by Gerko, and it was studied for non-finitely generated modules by Sharif.
		It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a ``long'' regular sequence in such a way that the quotient is still a normal affine monoid algebra.
		Abstract: Gerko (2001) has introduced the Cohen-Macaulay dimension for finitely generated modules over a local ring, and this homological dimension characterizes Cohen-Macaulay rings in a way one could hope for.
	www.math.unl.edu /~siyengar2/Seminars/Unlcas.html (1006 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Then a finitely generated $R$-module $M$ is said to {\it lift\/} to $Q$ if there exists a finitely generated $Q$-module $M'$ such that $x$ is $M'$-regular and $M \cong M'/xM'$.
		In this paper we give a general construction of finitely generated $R$-modules of finite projective dimension over $R$ which fail to lift to $Q$ provided $x \in \n^2$ and the depth of $R$ is at least 2.
		It is known that the even and odd Betti sequences of $M$ are eventually given by polynomials of the same degree $n$; the complexity of $M$ is the non-negative integer $n+1$.
	www.ma.utexas.edu /users/jorgen/papers.dir/papers.html (554 words)

	Projective Modules over the Endomorphism Ring of a Biuniform Module (Site not responding. Last check: 2007-10-08)
		Here a module M is biuniform if every two nonzero submodules of M has a nonzero intersection, and the sum of two proper submodules of M is proper.
		Dung and Facchini showed that every finitely generated projective module over the endomorphism ring of a biuniform module is free.
		We give examples of a uniserial module M, such that the endomorphism ring of M is a distributive ring do not admitting classical localization.
	www.math.ohiou.edu /~slopez/gena.html (183 words)

	Algebra Prelim, Spring 1993 (Site not responding. Last check: 2007-10-08)
		Assume that M is a nonzero finitely generated R-module with the property that the intersection of any two nonzero submodules is nonzero.
		Prove that if every nontrivial finite field extension of the field F has degree divisible by p, then every finite field extension of F has degree a power of p.
		Let D be a finite dihedral group, and let V be a finite dimensional complex vector space which is a D-module.
	www.math.vt.edu /people/linnell/Teaching/Algprelims/Spring93 (276 words)

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