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In
abstract algebra, a '''module''' is a generalization of a
vector space. In a vector space the set of
scalars forms a
field whereas in a module the scalars just form a
ring. 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.
== Definition ==
Specifically, a
left module over the ring ''R'' consists of an
abelian group (''M'', +) and an operation ''R'' × ''M''
-> ''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
''R''''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
right ''R''-module ''M'' or ''M''
''R'' is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' × ''R''
-> ''M'', and the above three axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''.
If ''R'' is
commutative, then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
== Examples ==
*If ''K'' is a
field, then the concepts "''K''-
vector space" and ''K''-module are identical.
*Every abelian group ''M'' is a module over the ring of
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''
''n'' 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 numbers form a ring ''R''. The set of all smooth
vector fields defined on ''X'' form a module over ''R'', and so do the
tensor fields and the
differential forms on ''X''.
*The square ''n''-by-''n''
matrices with real entries form a ring ''R'', and the
Euclidean space '''R'''
''n'' 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
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
map
''f'' : ''M''
-> ''N'' is a '''homomorphism of
R-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 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 theorems 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, written as ''R''-'''Mod'''. This is an
abelian category.
== Types of modules ==
'''Finitely generated.''' A module ''M'' is
finitely generated if there exist finitely many elements ''x''
1,...,''x''
''n'' 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 modules are
direct summands of free modules and share many of their desirable properties.
'''Injective.'''
Injective modules 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''. Simple modules are sometimes called ''irreducible''.
'''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.
'''Faithful.''' A
faithful module ''M'' is one where the action of each ''r'' in ''R'' gives an
injective map M→M. Equivalently, the
annihilator of ''M'' is the zero ideal.
'''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 endomorphism of the abelian group (''M'',+). The set of all group endomorphisms of ''M'' is denoted End
'''Z'''(''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
'''Z'''(''M'').
Such a ring homorphism ''R'' → End
'''Z'''(''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
'''Z'''(''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
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
''X'') and consider the
sheaves of O
''X''-modules. These form a category O
''X''-'''Mod'''. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O
''X''(''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
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