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Topic: Module homomorphism

Related Topics

Ring homomorphism

Commutative ring

Homomorphism

Algebraic structure

Algebra over a field

Polynomial

Symmetric bilinear form

Quadratic form

Homogeneous (mathematics)

Monomial

Quadratic function

Isomorphism

Exponential function

Algebra homomorphism

Chebyshev polynomials

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	Module (mathematics) - Wikipedia, the free encyclopedia
		Thus, a module, like a vector space, is an abelian group; a product is defined between elements of the ring and elements of the module, and this multiplication is associative (when used with the multiplication in the ring) and distributive.
		Modules of this type are called free and the number n is then the rank of the free module.
		An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules.
	en.wikipedia.org /wiki/Module_(mathematics) (1715 words)

	MC254 Algebra I
		Fundamental mathematical concepts developed in these modules, such as sets, functions and relations from MC144, and the number systems and polynomials and their properties from MC145, will be drawn upon to develop the theory of rings and modules in MC254.
		This module will provide students with a basic understanding of ring and module theory, one of the languages of advanced mathematics, which is employed in a number of fields of study, including group theory, number theory and linear algebra.
		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/Modules00-01/MC254.html (1142 words)

	Homomorphisms
		Suppose M is a matrix module over the coefficient ring R whose elements are a by b matrices and have domain D and codomain C. Suppose also that N is a matrix module over the coefficient ring R whose elements are a by c matrices and have domain D and codomain C'.
		Given a homomorphism a belonging to a submodule of Hom(M, N), and a homomorphism b belonging to a submodule of Hom(N, P), return the composition of the homomorphisms a and b as an element of Hom(M, P).
		The cokernel for the homomorphism a belonging to the module Hom(M, N).
	www.umich.edu /~gpcc/scs/magma/text787.htm (1723 words)

	Reference.com/Encyclopedia/Projective module
		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).
		Using the homomorphisms P → F and F → P for a projective module, it is easy to see that P has the same property; and also that if we can lift the identity P → P to P → F for F some free module mapping onto P, that P is a direct summand.
		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.
	www.reference.com /browse/wiki/Projective_module (1086 words)

	PlanetMath: decomposition of a module using orthogonal idempotents
		is not a homomorphism from the group into either the additive or multiplicative group of
		"decomposition of a module using orthogonal idempotents" is owned by alozano.
		This is version 6 of decomposition of a module using orthogonal idempotents, born on 2005-04-25, modified 2005-04-27.
	planetmath.org /encyclopedia/DecompositionOfAModuleUsingOrthogonalIdempotents.html (228 words)

	Module Homomorphisms (Site not responding. Last check: 2007-10-18)
		As you'd expect, a module homomorphism is a function f that maps one module into another, such that f commutes with group addition and scaling by r.
		The kernel of a module homomorphism is a submodule.
		If a module m is split into u and v, where u is unitary, any homomorphism on m can be restricted to u and v, and the resulting images lie in the corresponding submodules in the range.
	www.mathreference.com /mod,homo.html (402 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.
		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.
		The injective dimension of a A-module M is the infimum of lengths of an injective resolution of M.
	en.wikipedia.org /wiki/Injective_module (1043 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)

	GAP Manual: 41 Modules (Site not responding. Last check: 2007-10-18)
		That is, a module is stored either as a submodule of a module, or it is not (see Submodule, AsSubmodule for the details).
		Quotient modules Q = V / W of row modules are quotients of row spaces V, W that are both (row) modules for the same matrix algebra A.
		A free module of dimension n for an algebra A consists of all n-tuples of elements of A, the action of A is defined as component-wise multiplication from the right.
	www.math.jussieu.fr /~jmichel/htm/CHAP041.htm (1701 words)

	Homomorphisms
		A projective cover of a module M is a projective module P and a surjective homomorphism phi:P -> 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 n^(th) syzygy module of the module M. The module is constructed from the compact projective resolution of M. The compact resolution is constructed if it does not already exist.
		The projective resolutions of the first and second simple modules 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.
	magma.maths.usyd.edu.au /magma/htmlhelp/text883.htm (1489 words)

	[No title]
		But this is just the added requirement needed for a group homomorphism to be a module homomorphism.
		is a natural transformation by the properties of module homomorphisms (commuting with scalars).
		are module homomorphisms, the distributive laws are satisfied, and the other ring axioms are clear.
	www.math.washington.edu /~smith/Teaching/513nag/hwksols/hwksols.html (1691 words)

	PlanetMath: dual module
		is a module homomorphism, so is an element of
		Cross-references: function, fixed, bilinear form, isomorphic, commutative, action, homomorphisms, module, right, ring
		This is version 7 of dual module, born on 2006-06-14, modified 2006-10-24.
	planetmath.org /encyclopedia/DualModule.html (59 words)

	Tensor Product
		In general, a left r module and a right r module combine to form an abelian group, which is their tensor product.
		It follows that the tensor of finitely generated modules is finitely generated, and the tensor of finitely presented modules is finitely presented.
		This produces relations of the form (d(xy))z-(xy)(dz) We are interpreting t is an r module, hence d(xy) is the same as dx cross y or x cross dy.
	www.mathreference.com /mod-pit,tensor.html (2503 words)

	[ref] 67 The MeatAxe
		is guaranteed to be the same for the induced modules, but to obtain the complete relation to the original module, the bases used are needed as well.
		They are assumed to be modules over the same algebra so, in particular, they should have the same number of generators.
		It requires that one of the modules is known to be irreducible.
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/doc/htm/ref/CHAP067.htm (1001 words)

	Modules over a Matrix Algebra
		We construct a submodule of the permutation module for L(3, 4) in its representation of degree 21.
		The factors corresponding to the terms of a socle series for the A-module M. The factors are returned in the form of a sequence of A-modules in the order determined by a socle series for M. If M is irreducible, the function returns a sequence containing M alone.
		Given an A-module M, return a sequence Q of indecomposable summands of M. Each element of Q is an indecomposable submodule of M and M is equal to the (direct) sum of the terms of Q. If M is indecomposable, the sequence Q consists of M alone.
	www.math.lsu.edu /magma/text917.htm (4794 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)

	[No title]
		Suppose (:A(B is a ring homomorphism so that some element s (S is a zero divisor in A and ((s) (B is invertible.
		In case the homomorphism is a localization iS:A(S-1A, one can say more about the contraction operation, and, in addition, the extension operation has many good properties.
		(We point out that S-1A modules are also A-modules, via the ring homomorphism iS:A(S-1A.) The definition of S-1M, as well as the sum operation and scalar multiplication by elements of S-1A, and the proofs that these operations are well-defined, proceeds identically, symbol by symbol, with the construction of S-1A and the ring operations in S-1A.
	www.stanford.edu /class/math210b/LOCALIZATION.doc (1350 words)

	GAP Manual: 41.17. Module Homomorphisms (Site not responding. Last check: 2007-10-18)
		Let M_1 and M_2 be modules acted on by the rings R_1 and R_2 (via exponentiation), and varphi a ring homomorphism from R_1 to R_2.
		Suppose you have the module M_1 for the algebra R_1.
		If M_1 is a row module this is done by using the knowledge of images of a basis, if M_1 is a (quotient of a) free module then the algebra homomorphism and images of the generators of M_1 are used.
	www.math.uiuc.edu /Software/GAP-Manual/Module_Homomorphisms.html (209 words)

	[No title]
		M; each such homomorphism may be thought of as a candidate for the values of all operations on a typical elem* *ent of Mk.
		Th* en the homomorphisms* Q() and Q(ffl) in (6.27) and (6.28) are multiplicative and r* *espect the unit element.
		In fact, the three mo* st natural definitions are for stable modules, additively unstable modules, and unstable a *lgebras.
	www.math.purdue.edu /research/atopology/Boardman-Johnson-Wilson/bjw.txt (11687 words)

	cokernel -- cokernel of a map (Site not responding. Last check: 2007-10-18)
		-- produces the cokernel of the module homomorphism f
		The result will be a quotient module of the target of f.
		If f is a ring element, it is interpreted as a one by one matrix.
	www.stanford.edu /~mluciano/M2-help/0851.html (50 words)

	homomorphism - OneLook Dictionary Search
		Homomorphism : Online Plain Text English Dictionary [home, info]
		Homomorphism : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include homomorphism: lattice homomorphism, connecting homomorphism, frobenius homomorphism, graph homomorphism, module homomorphism, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=homomorphism (180 words)

	Modules
		The left module structure over opposite ring is defined by the same action, that is,
		The quotient module is the quotient Abelian group
		Notice that the action (1.23) is well defined.
	www.maths.warwick.ac.uk /~rumynin/rings2002/notes/node12.html (96 words)

	GAP Manual: 41.19 Module Homomorphism Records (Site not responding. Last check: 2007-10-18)
		the source of the homomorphism, a module M_1,
		the range of the homomorphism, a module M_2,
		the underlying algebra homomorphism from the ring acting on M_1 to the ring acting on M_2.
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C041S019.htm (110 words)

	Tensor Products of Modules
		Let K be a left A-module, M a left R-module.
		After this one is ready to show that is a homomorphism of algebras.
		Definition 4..12 A diagonal of an algebra A is a homomorphism
	www.maths.warwick.ac.uk /~rumynin/rings2002/ln/node34.html (280 words)

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