|
|
| Line 3: |
Line 3: |
| For modules in the [[Linux]] operating system kernel, see [[module (linux)]]. | | For modules in the [[Linux]] operating system kernel, see [[module (linux)]]. |
|
| |
|
| ----
| | For modules in [[abstract algebra]], see [[module (mathematics)]]. |
| | |
| In [[abstract algebra]], '''modules''' are generalizations of [[vector space]]s to arbitrary [[ring (mathematics)|ring]] of [[scalar]]s. Much of the theory of modules consists of recovering desirable properties of vector spaces in the realm of modules over certain rings. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a [[basis (linear algebra)|basis]].
| |
| | |
| == Definition ==
| |
| | |
| Specifically, a <b>left module</b> over the ring ''R'' consists of an [[abelian group]] (''M'', +) and an operation ''R'' × ''M'' <tt>-></tt> ''M'' (scalar multiplication, usually just written by juxtaposition, i.e. as ''rx'' for ''r'' in ''R'' and ''x'' in ''M'') such that
| |
| | |
| For all ''r'',''s'' in ''R'', ''x'',''y'' in ''M'', we have
| |
| # (''rs'')''x'' = ''r''(''sx'')
| |
| # (''r''+''s'')''x'' = ''rx''+''sx''
| |
| # ''r''(''x''+''y'') = ''rx''+''ry''
| |
| # 1''x'' = ''x''
| |
| | |
| Usually, we simply write "a left ''R''-module ''M''" or <sub>''R''</sub>''M''.
| |
| | |
| Some authors omit condition 4 for the general definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital, and all rings are assumed to have a one.
| |
| | |
| A <b>right ''R''-module</b> ''M'' or ''M''<sub>''R''</sub> is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' × ''R'' <tt>-></tt> ''M'', and the above three axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''.
| |
| | |
| If ''R'' is [[commutative ring|commutative]], then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
| |
| | |
| == Examples ==
| |
| | |
| *If ''K'' is a [[field (mathematics)|field]], then the concepts "''K''-[[vector space]]" and ''K''-module are identical.
| |
| *Every abelian group ''M'' is a module over the ring of [[integer|integers]] '''Z''' if we define ''nx'' = ''x'' + ''x'' + ... + ''x'' (''n'' summands) for ''n'' > 0, 0''x'' = 0, and (-''n'')''x'' = -(''nx'') for ''n'' < 0.
| |
| *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. The case ''n''=0 yields the trivial ''R''-module {0} consisting only of its identity element.
| |
| *If ''X'' is a smooth [[manifold]], then the smooth functions from ''X'' to the [[real number|real numbers]] form a ring ''R''. The set of all smooth [[vector field]]s defined on ''X'' form a module over ''R'', and so do the [[tensor field]]s and the [[differential form]]s on ''X''.
| |
| *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]].
| |
| *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.
| |
| | |
| == Submodules and homomorphisms ==
| |
| | |
| Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]]
| |
| 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).
| |
| | |
| If ''M'' and ''N'' are left ''R''-modules, then a [[function|map]]
| |
| ''f'' : ''M'' <tt>-></tt> ''N'' is a '''homomorphism of <i>R</i>-modules''' if, for any ''m, n'' in ''M''
| |
| and ''r, s'' in ''R'',
| |
| :''f''(''rm'' + ''sn'') = ''rf''(''m'') + ''sf''(''n'').
| |
| This, like any [[homomorphism]] of mathematical
| |
| objects, is just a mapping which preserves the structure of the objects.
| |
| | |
| 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.
| |
| | |
| The [[kernel (algebra)|kernel]] of a module homomorphism ''f'' : ''M'' <tt>-></tt> ''N'' is the submodule of ''M'' consisting of all elements that are sent to zero by ''f''. The [[isomorphism theorem]]s familiar from abelian groups and vector spaces are also valid for ''R''-modules.
| |
| | |
| The left ''R''-modules, together with their module homomorphisms, form a [[category theory|category]], written as ''R''-'''Mod'''. This is an [[abelian category]].
| |
| | |
| == Types of modules ==
| |
| | |
| '''Finitely generated.''' A 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 scalar ring ''R''.
| |
| | |
| '''Free.''' 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''. These are the modules that behave very much like vector spaces.
| |
| | |
| '''Projective.''' [[Projective module]]s are [[direct summand]]s of free modules and share many of their desirable properties.
| |
| | |
| '''Injective.''' [[Injective module]]s are defined dually to projective modules.
| |
| | |
| '''Simple.''' A [[simple module]] ''S'' is a module that is not {0} and whose only submodules are {0} and ''S''.
| |
| | |
| '''Indecomposable.''' An [[indecomposable module]] is a non-zero module that cannot be written as a [[direct sum]] of two non-zero submodules. Every simple module is indecomposable.
| |
| | |
| '''Noetherian.''' A [[noetherian module]] is a module whose every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.
| |
| | |
| '''Artinian.''' An [[artinian module]] is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.
| |
| | |
| == Alternative definition as representations ==
| |
| | |
| 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'').
| |
| | |
| Such a ring homorphism ''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.
| |
| | |
| 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|integers]] or over some [[modular arithmetic]] '''Z'''/''n'''''Z'''.
| |
| | |
| == Generalizations ==
| |
| | |
| 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'''.
| |
| | |
| Modules over ''commutative'' rings can be generalized in a different direction: take a [[ringed space]] (''X'', O<sub>''X''</sub>) and consider the [[sheaf|sheaves]] of O<sub>''X''</sub>-modules. These form a category O<sub>''X''</sub>-'''Mod'''. 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'').
| |
| | |
| ==References==
| |
| * F.W. Anderson and K.R. Fuller: ''Rings and Categories of Modules'', Graduate Texts in Mathematics, Vol. 13, 2 nd Ed., Springer-Verlag, New York, 1992
| |
| | |
| <!-- Interlanguage links -->
| |
| [[de:Modul]][[pl:Modu%C5%82 (matematyka)]][[zh:模]]
| |
Generally, a module is a component of a system that has a well-defined interface to the other components; something is modular if it is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair of the components.
For modules in the Linux operating system kernel, see module (linux).
For modules in abstract algebra, see module (mathematics).