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Topic: Projective module

Related Topics

Injective module

Cochain complex

Exact functor

Category theory

Cohomology

Homological algebra

Samuel Eilenberg

Topology

Saunders Mac Lane

Norman Steenrod

September 30

	Springer Online Reference Works
		Kaplansky's theorem [2], asserting that every projective module is a direct sum of projective modules with countably many generators, reduces the study of the structure of projective modules to the countable case.
		Projective modules with finitely many generators are studied in algebraic
		The coincidence of the class of projective modules and that of free modules has been proved for local rings [2], and for rings of polynomials in several variables over a field (see [3], [4]).
	eom.springer.de /p/p075280.htm (454 words)

	Projective module - Wikipedia, the free encyclopedia
		In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors).
		Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
		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.
	en.wikipedia.org /wiki/Projective_module (1057 words)

	Projective Module -- from Wolfram MathWorld
		A projective module generalizes the concept of the free module.
		The notion of projective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors.
		It is dual to the notion of injective module.
	mathworld.wolfram.com /ProjectiveModule.html (262 words)

	Homomorphisms
		The space of projective homomorphisms from module M to module N. That is, the space of all homomorphisms that factor through a projective module.
		A projective cover of a module M is a projective module P and a surjective homomorphism phi:P longrightarrow M such that P is minimal with respect to the property of having such a surjective homomorphism to M. A projective resolution to n steps of an A-module M is a pair consisting of a complex
		The projective resolution of the first and second simple module appear to have exponential rates of growth but the terms after the second term are all direct sums of copies of the third projective module.
	www.umich.edu /~gpcc/scs/magma/text1008.htm (1383 words)

	Module (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring.
		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.
		Projective modules are direct summands of free modules and share many of their desirable properties.
	en.wikipedia.org /wiki/Module_(mathematics) (1708 words)

	Modules over Basic Algebras
		A module M over a basic algebra B is presented as a sequence of matrices, one for each generator of the algebra.
		The indecomposable projective modules are defined from the structure of the algebra and have associated path trees that solve the homomorphism lifting problem.
		The module is the quotient of the i^(th) projective module by its radical.
	www.umich.edu /~gpcc/scs/magma/text1007.htm (774 words)

	Opposite Algebras
		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.
	www.math.lsu.edu /magma/text858.htm (747 words)

	PlanetMath: projective module
		is a direct summand of a free module.
		Cross-references: free module, direct summand, epimorphism, functor, short exact sequence, equivalent, satisfies, module
		This is version 3 of projective module, born on 2002-01-05, modified 2003-09-20.
	planetmath.org /encyclopedia/ProjectiveModule.html (100 words)

	Projective and Injective Modules (Site not responding. Last check: 2007-09-19)
		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)

	Projective Modules over the Endomorphism Ring of a Biuniform Module (Site not responding. Last check: 2007-09-19)
		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 /~lopez/gena.html (183 words)

	[No title]
		Notice that the ring R (2, 3, 2) has all the good module theoretic properties that could be desired; since every full matrix is regular, the flat modules are all spacial, and it is known that the left and right global dimension is 2.
		Recall that every finitely generated projective left R-module is the homomorphic image of an idempotent matrix (viewed as an endomorphism of a finitely generated free left R-module) and that the image of every idempotent matrix is a finitely generated projective left R-module.
		In fact from the connection between finitely generated projective left modules and idempotent matrices, it is clear that (30)-(32) are precisely the conditions for r to determine a projective rank function.
	www.math.rutgers.edu /~sontag/FTP_DIR/wdicks.txt (11071 words)

	[No title]
		The pure projective dimension FAILURE OF BROWN REPRESENTABILITY 5 of an R-module M is defined to be the length of its shortest pure resolution by pure projectives.
		The pure injective dimension of a module I is the length of the shortest pure resolution by pure injectives.
		The integer j and the modules M and N are clearly determined by the homology of F.
	jdc.math.uwo.ca /papers/purity.txt (7106 words)

	Springer Online Reference Works
		Free module) is defined as the number of its free generators.
		In general, the rank of a free module is not uniquely defined.
		In this case the concept of the rank of a module can be extended to projective modules as follows.
	eom.springer.de /R/r077470.htm (211 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		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.
		A is not noetherian, so there exists an ideal J that is not of finite type.
		But, if A/J was projective, the following exact sequence would split : 0 -> J -> A -> A/J -> 0 It would mean that J is generated by one element, contradiction.
	www.math.niu.edu /~rusin/known-math/01_incoming/flat_module (259 words)

	Group algebras of p-groups
		Creates the data on the chain maps for all generators of the cohomology of the simple module k in degrees within the limits of the compact projective resolution PR of the simple module.
		The input can be given either as the module k and the number of steps n or as the compact projective resolution PR of k together with AC, the calculation of the chain map generators of the cohomology.
		Computes the chain map from the resolution P2 of the simple module for the basic algebra of a subgroup H of a group G to the restriction to H of the resolution P1 of the simple module for the basic algebra of G. The inputs P1 and P2 must be in compact form.
	www.math.lsu.edu /magma/text860.htm (1170 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		I'm not sure I've heard of the concept of "homological dimension of a module"; more precisely, if I heard it, I would translate it into "projective dimension".
		The latter is simply the length of the shortest projective resolution of the module.
		You're not likely to get good examples of the type you describe, since if M is a Z[G]-module, it can only have finite projective dimension if that dimension is at most 1, that is, M is either projective or the quotient of a projective module by a projective submodule.
	www.math.niu.edu /~rusin/known-math/97/proj.dim (156 words)

	[No title]
		The answer is, mutatis mutandis, always the same: modules over a `ring' are * characterized by the existence of a `small generator', which plays the role of the free modul *e of rank one.
		Fo* r modules* over a ring, smallness is closely related to finite generation: every finitely * generated module* is small and for projective modules, `small' and `finitely generated' are equiv* *alent concepts.
		This uses that P is projective and so that A(P, -) is an e* xact functor and both sides of the map (2.6)are right exact in X. Since P is a generator, every object can be written as the cokernel of a morphi *sm between sums of copies of P.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	PlanetMath: finitely generated projective module
		"finitely generated projective module" is owned by mhale.
		Cross-references: inner product, projection, unital, idempotent, right, finitely generated, unital ring
		This is version 3 of finitely generated projective module, born on 2003-02-26, modified 2003-11-18.
	planetmath.org /encyclopedia/FiniteProjectiveModule.html (58 words)

	AMCA: Comparing the projective modules of higher Frobenius kernels and finite Chevalley groups by Zongzhu Lin (Site not responding. Last check: 2007-09-19)
		Question is answered by Lin and Nakano using a May spectral sequence and Quillen's isomorphism for p-groups for all p which is larger than the edge multiplicities of its Dynkin diagram.
		Instead of using the Quillen's isomorphism of associated graded algebra of the group algebra a Sylow subgroup to the restricted enveloping algebra of the Lie algebra of the unipotent group, the question is reduced to the Borel subgroup B. We conjecture that if M is a B module that projective as a B
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/o/h/33.htm (337 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)

	Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, Vol. 46, No. 2, pp. 447-466, 2005 (Site not responding. Last check: 2007-09-19)
		An $R$-module $M$ is a GGCD module if $M$ is multiplication and the set of finitely generated faithful multiplication submodules of $M$ is closed under intersection.
		As a generalization of Glaz GGCD ring we say that an $R$-module $M$ is a Glaz GGCD module if $M$ is finitely generated faithful multiplication, every cyclic submodule of $M$ is projective, and the set of finitely generated projective (flat) submodules of $M$ is closed under intersection.
		Various properties and characterizations of GGCD modules and Glaz GGCD modules are considered.
	www.emis.de /journals/BAG/vol.46/no.2/12.html (215 words)

	AMCA: Modules Having -Radical by Ayse Cigdem Ozcan (Site not responding. Last check: 2007-09-19)*
		In this note we characterize rings in terms of modules that has *-radical.
		After that we show that R is a right H-ring if and only if every right R-module that has -radical is lifting and, R is a semiprimary QF-3 ring if and only if R is right perfect and every projective* right R-module that has *-radical is injective (extending).
		(R)=Z(R) (Z(R) is the singular right ideal of R), R is semiperfect and every projective right R-module that has *-radical is injective (extending).
	at.yorku.ca /c/a/b/w/08.htm (209 words)

	Mrinal Kanti Das (Site not responding. Last check: 2007-09-19)
		The Euler class group of a polynomial algebra.
		The Euler class groups of polynomial rings and unimodular elements in projective modules (with Raja Sridharan).
		Gave two lectures on ``Unimodular elements in projective modules'' in the Basic Notions Seminar.
	www.mri.ernet.in /~annlrepo/node6.html (187 words)

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