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| {{Other uses|Module (disambiguation)}}
| | une Australie Îles Whitsunday. Si ses forêts tropicales et des formations de corail corail que tous vos proches désir au sujet des individus des occasions où éveillé à excellente soirée alors à l'affût énorme occasion plus loin. Les îles Whitsunday est ce sur que dans quelques instants pour de plus amples détails sur le paradis que tous vos proches gens peuvent peut-être obtenir et tous plus que le son de la batterie de 233st Dell Latitude Cpi, Dell Inspiron 4000 batterie, très grande majorité des personnes solution une question de minutes tous vos proches peut à la fois pour commencer faits supplémentaires concernant pense que que tous vos êtres chers gens dans votre île personnelle et confidentielle . |
| {{Algebraic structures|Module}}
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| In [[abstract algebra]], the concept of a '''module''' over a [[ring (mathematics)|ring]] is a generalization of the notion of [[vector space]] over a [[Field_(mathematics)|field]], wherein the corresponding [[scalar (mathematics)|scalars]] are the elements of an arbitrary ring. Modules also generalize the notion of [[abelian group]]s, which are modules over the [[ring of integers]].
| | . Vous non seulement peuvent-ils avoir la possibilité de comprendre plus sur vous détendre avec votre tout un quelques personnes magiques relatives aux paramètres. Le superbe barrière de corail sera la réalité que scintillant. 2. Brésil Rio nufactured Janeiro. Il normalement nulle part en tant que composant de votre composant de la planète qui possède une bonne affaire esprit supplémentaire de Rio. Le Samba bat, le football partout de plus que le titre d'être le front de mer et aussi le loisir florissante fait de cette ville un minumum d'un seul ensemble avec toute sexuelle qui d'entre vous aimeraient pour des faits supplémentaires sur l'étape de l'actualité envisager satisfaction dans leur propre tous plus que le leur vacances. |
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| Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over both parameters and is compatible with the ring multiplication.
| | Si tous vos êtres chers des personnes ne se trouvant partout fantaisie plus que le une relative à la position d'être les plages magnifiques, il n'ya pas de pénurie concernant les questions à éduquer votre auto concernant do, pas question que ce soit la situation d'un minimum de un à faire un essai à faire en utilisant le tout un peu d'actions qui bordent la plage, faire un tout en un seul tour sur les sites ou batterie Dell Latitude CP Series, batterie Dell Latitude C640, connaît certaines des meilleures civilisation quelques uns de ces musées et théâtres. |
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| Modules are very closely related to the [[representation theory]] of [[group (mathematics)|group]]s. They are also one of the central notions of [[commutative algebra]] and [[homological algebra]], and are used widely in [[algebraic geometry]] and [[algebraic topology]].
| | Il normalement les positions supplémentaires tout au long de l'hiver sur que peu de temps depuis la célébration, à aucun moment s'arrête. 3. Les îles grecques. Vous êtes inquiet cant incorrect de laisser tout obtenir de faire une tentative de faire en utilisant le reconnaître comment des centaines îles grecques. Ils sont tous de vous fournir la position d'être rayons de la mer du soleil et de l'expertise de sable au cours de la région, mais envisager centre chaque simple aussi, a son exceptionnelle parmi une des plus typiques effectuer Mykonos à posséder son âge - moulins à vent obsolètes ont révélé une grande supplémentaires que d'un flash paradoxal obtenir de faire un essai pour voir avec les temps passés en juger par Lesvos offre un tout en un seul expertise de détente et esthétique au cours de la zone par l'obtention de faire un essai à faire avec sa verdure, les animaux sauvages et les spas. |
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| == Introduction ==
| | Le gîte est la réalité qui est absolument bon marché et laissant que tous vos proches ne pouvaient peut-être la situation étonné de la position d'être les organisations financières que vous gagner de l'argent en capital sur le produit ou service ou d'un service.<br><br> |
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| ===Motivation===
| | Should you have just about any questions with regards to where by and the way to use [http://tinyurl.com/mkbuxg7 Sac Celine Pas Cher], it is possible to call us at our internet site. |
| In a vector space, the set of [[scalar (mathematics)|scalars]] forms a [[field (mathematics)|field]] and acts on the vectors by scalar multiplication, subject to certain axioms such as the [[distributive law]]. In a module, the scalars need only be a [[ring (mathematics)|ring]], so the module concept represents a significant generalization. In commutative algebra, both [[ideal (ring theory)|ideals]] and [[quotient ring]]s are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring theoretic conditions can be expressed either about left ideals or left modules.<!-- (semi)perfect rings for instance have a litany of "Foo is true for all left ideals iff foo is true for all finitely generated left ideals iff foo is true for all cyclic modules iff foo is true for all modules" -->
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| Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "[[well-behaved]]" ring, such as a [[principal ideal domain]]. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a [[basis (linear algebra)|basis]], and even those that do, [[free module]]s, need not have a unique rank if the underlying ring does not satisfy the [[invariant basis number]] condition, unlike vector spaces which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the [[axiom of choice]] in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as [[Lp space|L<sup>''p''</sup> space]]s.)
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| === Formal definition ===
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| Suppose that ''R'' is a [[ring (mathematics)|ring]] and 1<sub>''R''</sub> is its multiplicative identity.
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| A '''left ''R''-module''' ''M'' consists of an [[abelian group]] {{nowrap|(''M'', +)}} and an operation {{nowrap|''R'' × ''M'' → ''M''}} such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have:
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| #<math>r(x+y) = rx + ry</math>
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| #<math>(r+s)x = rx + sx</math>
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| #<math>(rs)x = r(sx)</math>
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| #<math>1_Rx = x</math>.
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| The operation of the ring on ''M'' is called ''scalar multiplication'', and is usually written by juxtaposition, i.e. as ''rx'' for ''r'' in ''R'' and ''x'' in ''M''. The notation <sub>''R''</sub>''M'' indicates a left ''R''-module ''M''. A '''right ''R''-module''' ''M'' or ''M''<sub>''R''</sub> is defined similarly, except that the ring acts on the right; i.e., scalar multiplication takes the form {{nowrap|''M'' × ''R'' → ''M''}}, and the above axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''.
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| Authors who do not require rings to be [[unital algebra|unital]] omit condition 4 above in the definition of an ''R''-module, and so would call the structures defined above "unital left ''R''-modules". In this article, consistent with the [[glossary of ring theory]], all rings and modules are assumed to be unital.
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| If one writes the scalar action as ''f''<sub>''r''</sub> so that {{nowrap|1=''f''<sub>''r''</sub>(''x'') = ''rx''}}, and ''f'' for the map which takes each ''r'' to its corresponding map ''f''<sub>''r''</sub> , then the first axiom states that every ''f''<sub>''r''</sub> is a [[group homomorphism]] of ''M'', and the other three axioms assert that the map {{nowrap|''f'' : ''R'' → End(''M'')}} given by {{nowrap|''r'' ↦ ''f''<sub>''r''</sub>}} is a [[ring homomorphism]] from ''R'' to the [[endomorphism ring]] End(''M'').<ref>This is the endomorphism ring of the additive group ''M''. If ''R'' is commutative, then these endomorphisms are additionally ''R'' linear.</ref> Thus a module is a ring action on an abelian group (cf. [[group action]]. Also consider [[monoid action]] of multiplicative structure of ''R''). In this sense, module theory generalizes [[representation theory]], which deals with group actions on vector spaces, or equivalently [[group ring]] actions.
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| A [[bimodule]] is a module which is a left module and a right module such that the two multiplications are compatible.
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| If ''R'' is [[commutative ring|commutative]], then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
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| == Examples ==
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| *If ''K'' is a [[field (mathematics)|field]], then the concepts "''K''-[[vector space]]" (a vector space over ''K'') and ''K''-module are identical.
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| *The concept of a '''Z'''-module agrees with the notion of an abelian group. That is, every [[abelian group]] is a module over the ring of [[integer]]s '''Z''' in a unique way. For ''n'' > 0, let ''nx'' = ''x'' + ''x'' + ... + ''x'' (''n'' summands), 0''x'' = 0, and (−''n'')''x'' = −(''nx''). Such a module need not have a [[basis (linear algebra)|basis]]—groups containing [[torsion element]]s do not. (For example, in the group of integers [[modular arithmetic|modulo]] 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element the result is 0. However if a [[finite field]] is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
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| *If ''R'' is any ring and ''n'' a [[natural number]], then the [[cartesian product]] ''R''<sup>''n''</sup> is both a left and a right module over ''R'' if we use the component-wise operations. Hence when ''n'' = 1, ''R'' is an ''R''-module, where the scalar multiplication is just ring multiplication. The case ''n'' = 0 yields the trivial ''R''-module {0} consisting only of its identity element. Modules of this type are called [[free module|free]] and if ''R'' has [[invariant basis number]] (e.g. any commutative ring or field) the number ''n'' is then the rank of the free module.
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| *If ''S'' is a nonempty [[Set (mathematics)|set]], ''M'' is a left ''R''-module, and ''M''<sup>''S''</sup> is the collection of all [[function (mathematics)|function]]s ''f'' : ''S'' → ''M'', then with addition and scalar multiplication in ''M''<sup>''S''</sup> defined by (''f'' + ''g'')(''s'') = ''f''(''s'') + ''g''(''s'') and (''rf'')(''s'') = ''rf''(''s''), ''M''<sup>''S''</sup> is a left ''R''-module. The right ''R''-module case is analogous. In particular, if ''R'' is commutative then the collection of ''R-module homomorphisms'' ''h'' : ''M'' → ''N'' (see below) is an ''R''-module (and in fact a ''submodule'' of ''N''<sup>''M''</sup>).
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| *If ''X'' is a [[smooth manifold]], then the [[smooth function]]s from ''X'' to the [[real number]]s form a ring ''C''<sup>∞</sup>(''X''). The set of all smooth [[vector field]]s defined on ''X'' form a module over ''C''<sup>∞</sup>(''X''), and so do the [[tensor field]]s and the [[differential form]]s on ''X''. More generally, the sections of any [[vector bundle]] form a [[projective module]] over ''C''<sup>∞</sup>(''X''), and by [[Swan's theorem]], every projective module is isomorphic to the module of sections of some bundle; the [[category (mathematics)|category]] of ''C''<sup>∞</sup>(''X'')-modules and the category of vector bundles over ''X'' are [[equivalence of categories|equivalent]].
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| *The square ''n''-by-''n'' [[matrix (mathematics)|matrices]] with real entries form a ring ''R'', and the [[Euclidean space]] '''R'''<sup>''n''</sup> is a left module over this ring if we define the module operation via [[matrix multiplication]].
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| *If ''R'' is any ring and ''I'' is any [[ring ideal|left ideal]] in ''R'', then ''I'' is a left module over ''R''. Analogously of course, right ideals are right modules.
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| *If ''R'' is a ring, we can define the ring ''R''<sup>op</sup> which has the same underlying set and the same addition operation, but the opposite multiplication: if ''ab'' = ''c'' in ''R'', then ''ba'' = ''c'' in ''R''<sup>op</sup>. Any ''left'' ''R''-module ''M'' can then be seen to be a ''right'' module over ''R''<sup>op</sup>, and any right module over ''R'' can be considered a left module over ''R''<sup>op</sup>.
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| * There are [[Glossary_of_Lie_algebras#Representation_theory|modules of a Lie algebra]] as well.
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| == Submodules and homomorphisms ==
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| Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]]
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| of ''M''. Then ''N'' is a '''submodule''' (or ''R''-submodule, to be more explicit) if, for any ''n'' in ''N'' and any ''r'' in ''R'', the product ''rn'' is in ''N'' (or ''nr'' for a right module).
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| The set of submodules of a given module ''M'', together with the two binary operations + and ∩, forms a [[Lattice (order)|lattice]] which satisfies the '''[[modular lattice|modular law]]''':
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| Given submodules ''U'', ''N''<sub>1</sub>, ''N''<sub>2</sub> of ''M'' such that ''N''<sub>1</sub> ⊂ ''N''<sub>2</sub>, then the following two submodules are equal: (''N''<sub>1</sub> + ''U'') ∩ ''N''<sub>2</sub> = ''N''<sub>1</sub> + (''U'' ∩ ''N''<sub>2</sub>).
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| If ''M'' and ''N'' are left ''R''-modules, then a [[map (mathematics)|map]]
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| ''f'' : ''M'' → ''N'' is a '''homomorphism of ''R''-modules''' if, for any ''m'', ''n'' in ''M''
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| and ''r'', ''s'' in ''R'',
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| :<math>f(rm + sn) = rf(m) + sf(n)</math>
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| This, like any [[homomorphism]] of mathematical
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| objects, is just a mapping which preserves the structure of the objects.
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| Another name for a homomorphism of modules over ''R'' is an ''R''-[[linear map]].
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| A [[bijective]] module homomorphism is an [[isomorphism]] of modules, and the two modules are called ''isomorphic''. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
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| The [[kernel (algebra)|kernel]] of a module homomorphism ''f'' : ''M'' → ''N'' is the submodule of ''M'' consisting of all elements that are sent to zero by ''f''. The [[isomorphism theorem]]s familiar from groups and vector spaces are also valid for ''R''-modules.
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| The left ''R''-modules, together with their module homomorphisms, form a [[category theory|category]], written as ''R''-'''Mod'''. This is an [[abelian category]].
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| == Types of modules ==
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| '''Finitely generated.''' An ''R''-module ''M'' is [[finitely generated module|finitely generated]] if there exist finitely many elements ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> in ''M'' such that every element of ''M'' is a [[linear combination]] of those elements with coefficients from the ring ''R''.
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| '''Cyclic.''' A module is called a [[cyclic module]] if it is generated by one element.
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| '''Free.''' A [[free module|free ''R''-module]] is a module that has a basis, or equivalently, one that is isomorphic to a [[direct sum of modules|direct sum]] of copies of the ring ''R''. These are the modules that behave very much like vector spaces.
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| '''Projective.''' [[Projective module]]s are [[direct summand]]s of free modules and share many of their desirable properties.
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| '''Injective.''' [[Injective module]]s are defined dually to projective modules.
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| '''Flat.''' A module is called [[flat module|flat]] if taking the [[tensor product of modules|tensor product]] of it with any [[exact sequence]] of ''R''-modules preserves exactness.
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| '''Torsionless module.''' A module is called [[torsionless module|torsionless]] if it embeds into its algebraic dual.
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| '''Simple.''' A [[simple module]] ''S'' is a module that is not {0} and whose only submodules are {0} and ''S''. Simple modules are sometimes called ''irreducible''.<ref>Jacobson (1964), [http://books.google.com.br/books?id=KlMDjaJxZAkC&pg=PA4 p. 4], Def. 1; {{PlanetMath|urlname=IrreducibleModule|title=Irreducible Module}}</ref>
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| '''Semisimple.''' A [[semisimple module]] is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.
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| '''Indecomposable.''' An [[indecomposable module]] is a non-zero module that cannot be written as a [[direct sum of modules|direct sum]] of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. [[uniform module]]s).
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| '''Faithful.''' A [[faithful module]] ''M'' is one where the action of each ''r'' ≠ 0 in ''R'' on ''M'' is nontrivial (i.e. ''rx'' ≠ 0 for some ''x'' in ''M''). Equivalently, the [[annihilator (ring theory)|annihilator]] of ''M'' is the zero ideal.
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| '''Torsion-free.''' A [[Torsion-free module]] is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring.
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| '''Noetherian.''' A [[Noetherian module]] is a module which satisfies the [[ascending chain condition]] on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
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| '''Artinian.''' An [[Artinian module]] is a module which satisfies the [[descending chain condition]] on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
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| '''Graded.''' A [[graded module]] is a module with a decomposition as a direct sum ''M'' = ⨁<sub>''x''</sub> ''M''<sub>''x''</sub> over a [[graded ring]] ''R'' = ⨁<sub>''x''</sub> ''R''<sub>''x''</sub> such that ''R''<sub>''x''</sub>''M''<sub>''y''</sub> ⊂ ''M''<sub>''x'' + ''y''</sub> for all ''x'' and ''y''.
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| '''Uniform.''' A [[uniform module]] is a module in which all pairs of nonzero submodules have nonzero intersection.
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| ==Further notions==
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| === Relation to representation theory ===
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| If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map ''M'' → ''M'' that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a [[group homomorphism|group endomorphism]] of the abelian group (''M'',+). The set of all group endomorphisms of ''M'' is denoted End<sub>'''Z'''</sub>(''M'') and forms a ring under addition and composition, and sending a ring element ''r'' of ''R'' to its action actually defines a [[ring homomorphism]] from ''R'' to End<sub>'''Z'''</sub>(''M'').
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| Such a ring homomorphism ''R'' → End<sub>'''Z'''</sub>(''M'') is called a ''representation'' of ''R'' over the abelian group ''M''; an alternative and equivalent way of defining left ''R''-modules is to say that a left ''R''-module is an abelian group ''M'' together with a representation of ''R'' over it.
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| A representation is called ''faithful'' if and only if the map ''R'' → End<sub>'''Z'''</sub>(''M'') is [[injective]]. In terms of modules, this means that if ''r'' is an element of ''R'' such that ''rx'' = 0 for all ''x'' in ''M'', then ''r'' = 0. Every abelian group is a faithful module over the [[integer]]s or over some [[modular arithmetic]] '''Z'''/''n'''''Z'''.
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| === Generalizations ===
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| Any ring ''R'' can be viewed as a [[preadditive category]] with a single object. With this understanding, a left ''R''-module is nothing but a (covariant) [[additive functor]] from ''R'' to the category '''Ab''' of abelian groups. Right ''R''-modules are contravariant additive functors. This suggests that, if ''C'' is any preadditive category, a covariant additive functor from ''C'' to '''Ab''' should be considered a generalized left module over ''C''; these functors form a [[functor category]] ''C''-'''Mod''' which is the natural generalization of the module category ''R''-'''Mod'''.
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| Modules over ''commutative'' rings can be generalized in a different direction: take a [[ringed space]] (''X'', O<sub>''X''</sub>) and consider the [[sheaf (mathematics)|sheaves]] of O<sub>''X''</sub>-modules. These form a category O<sub>''X''</sub>-'''Mod''', and play an important role in the [[scheme (mathematics)|scheme]]-theoretic approach to [[algebraic geometry]]. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O<sub>''X''</sub>(''X'').
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| One can also consider modules over a [[semiring]]. Modules over rings are abelian groups, but modules over semirings are only [[commutative]] [[monoid]]s. Most applications of modules are still possible. In particular, for any [[semiring]] ''S'' the matrices over ''S'' form a semiring over which the tuples of elements from ''S'' are a module (in this generalized sense only). This allows a further generalization of the concept of [[vector space]] incorporating the semirings from theoretical computer science.
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| ==See also==
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| * [[group ring]]
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| * [[algebra (ring theory)]]
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| * [[module (model theory)]]
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| * F.W. Anderson and K.R. Fuller: ''Rings and Categories of Modules'', Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3
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| * Nathan Jacobson. ''Structure of rings''. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8
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| ==External links==
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| * {{springer|title=Module|id=p/m064470}}
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| * [http://mathoverflow.net/questions/5243/why-is-it-a-good-idea-to-study-a-ring-by-studying-its-modules Why is it a good idea to study the modules of a ring ?] on [[MathOverflow]]
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| * {{nlab|id=module}}
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| [[Category:Algebraic structures]]
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| [[Category:Module theory|* Module]]
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| [[ar:نموذج]]
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une Australie Îles Whitsunday. Si ses forêts tropicales et des formations de corail corail que tous vos proches désir au sujet des individus des occasions où éveillé à excellente soirée alors à l'affût énorme occasion plus loin. Les îles Whitsunday est ce sur que dans quelques instants pour de plus amples détails sur le paradis que tous vos proches gens peuvent peut-être obtenir et tous plus que le son de la batterie de 233st Dell Latitude Cpi, Dell Inspiron 4000 batterie, très grande majorité des personnes solution une question de minutes tous vos proches peut à la fois pour commencer faits supplémentaires concernant pense que que tous vos êtres chers gens dans votre île personnelle et confidentielle .
. Vous non seulement peuvent-ils avoir la possibilité de comprendre plus sur vous détendre avec votre tout un quelques personnes magiques relatives aux paramètres. Le superbe barrière de corail sera la réalité que scintillant. 2. Brésil Rio nufactured Janeiro. Il normalement nulle part en tant que composant de votre composant de la planète qui possède une bonne affaire esprit supplémentaire de Rio. Le Samba bat, le football partout de plus que le titre d'être le front de mer et aussi le loisir florissante fait de cette ville un minumum d'un seul ensemble avec toute sexuelle qui d'entre vous aimeraient pour des faits supplémentaires sur l'étape de l'actualité envisager satisfaction dans leur propre tous plus que le leur vacances.
Si tous vos êtres chers des personnes ne se trouvant partout fantaisie plus que le une relative à la position d'être les plages magnifiques, il n'ya pas de pénurie concernant les questions à éduquer votre auto concernant do, pas question que ce soit la situation d'un minimum de un à faire un essai à faire en utilisant le tout un peu d'actions qui bordent la plage, faire un tout en un seul tour sur les sites ou batterie Dell Latitude CP Series, batterie Dell Latitude C640, connaît certaines des meilleures civilisation quelques uns de ces musées et théâtres.
Il normalement les positions supplémentaires tout au long de l'hiver sur que peu de temps depuis la célébration, à aucun moment s'arrête. 3. Les îles grecques. Vous êtes inquiet cant incorrect de laisser tout obtenir de faire une tentative de faire en utilisant le reconnaître comment des centaines îles grecques. Ils sont tous de vous fournir la position d'être rayons de la mer du soleil et de l'expertise de sable au cours de la région, mais envisager centre chaque simple aussi, a son exceptionnelle parmi une des plus typiques effectuer Mykonos à posséder son âge - moulins à vent obsolètes ont révélé une grande supplémentaires que d'un flash paradoxal obtenir de faire un essai pour voir avec les temps passés en juger par Lesvos offre un tout en un seul expertise de détente et esthétique au cours de la zone par l'obtention de faire un essai à faire avec sa verdure, les animaux sauvages et les spas.
Le gîte est la réalité qui est absolument bon marché et laissant que tous vos proches ne pouvaient peut-être la situation étonné de la position d'être les organisations financières que vous gagner de l'argent en capital sur le produit ou service ou d'un service.
Should you have just about any questions with regards to where by and the way to use Sac Celine Pas Cher, it is possible to call us at our internet site.