Module (mathematics): Difference between revisions

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In [[mathematics]], a '''module''' is a generalization of the notion of [[vector space]] in which the [[Field (mathematics)|field]] of [[scalar (mathematics)|scalars]] is replaced by a [[Ring (mathematics)|ring]]. The concept of ''module'' generalizes also the notion of [[abelian group]], since the abelian groups are exactly the modules over the ring of [[integer]]s.

Like a vector space, a module is an additive abelian group, and scalar multiplication is [[Distributive property|distributive]] over the operationoperations of addition between elements of the ring or module and is [[Semigroup action|compatible]] with the ring multiplication.

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]].

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In a vector space, the set of [[scalar (mathematics)|scalars]] is 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" -->

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible 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 for those that do ([[free module]]s) the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique [[Free_module#Definition|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]] vector spaces, or certain well-behaved infinite-dimensional vector spaces such as [[Lp space|L^''p'' space]]s.)

=== Formal definition ===

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*If ''K'' is a [[field (mathematics)|field]], then ''K''-[[vector space]]s (vector spaces over ''K'') and ''K''-modules are identical.

*If ''K'' is a field, and ''K''[''x''] a univariate [[polynomial ring]], then a [[Polynomial ring#Modules|''K''[''x'']-module]] ''M'' is a ''K''-module with an additional action of ''x'' on ''M'' by a group homomorphism that commutes with the action of ''K'' on ''M''. In other words, a ''K''[''x'']-module is a ''K''-vector space ''M'' combined with a [[linear map]] from ''M'' to ''M''. Applying the [[structure theorem for finitely generated modules over a principal ideal ___domain]] to this example shows the existence of the [[Rational canonical form|rational]] and [[Jordan normal form|Jordan canonical]] forms.

*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 {{nowrap|''n'' > 0}}, let {{nowrap|1=''n'' ⋅ ''x'' = ''x'' + ''x'' + ... + ''x''}} (''n'' summands), {{nowrap|1=0 ⋅ ''x'' = 0}}, and {{nowrap|1=(−''n'') ⋅ ''x'' = −(''n'' ⋅ ''x'')}}. 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 whichthat 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.)

*The [[decimal fractions]] (including negative ones) form a module over the integers. Only [[singleton (mathematics)|singletons]] are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.

*If ''R'' is any ring and ''n'' a [[natural number]], then the [[cartesian product]] ''R''^''n'' is both a left and right ''R''-module over ''R'' if we use the component-wise operations. Hence when {{nowrap|1=''n'' = 1}}, ''R'' is an ''R''-module, where the scalar multiplication is just ring multiplication. The case {{nowrap|1=''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.

*If M_''n''(''R'') is the ring of {{nowrap|''n''&thinsp;×&thinsp;''n''}} [[matrix (mathematics)|matrices]] over a ring ''R'', ''M'' is an M_''n''(''R'')-module, and ''e''_''i'' is the {{nowrap|''n'' × ''n''}} matrix with 1 in the {{nowrap|(''i'', ''i'')}}-entry (and zeros elsewhere), then ''e''_''i''''M'' is an ''R''-module, since {{nowrap|1=''re''_''i''''m'' = ''e''_''i''''rm'' ∈ ''e''_''i''''M''}}. So ''M'' breaks up as the [[direct sum]] of ''R''-modules, {{nowrap|1=''M'' = ''e''₁''M'' ⊕ ... ⊕ ''e''_''n''''M''}}. Conversely, given an ''R''-module ''M''₀, then ''M''₀^⊕''n'' is an M_''n''(''R'')-module. In fact, the [[category of modules|category of ''R''-modules]] and the [[category (mathematics)|category]] of M_''n''(''R'')-modules are [[equivalence of categories|equivalent]]. The special case is that the module ''M'' is just ''R'' as a module over itself, then ''R''^''n'' is an M_''n''(''R'')-module.

*If ''S'' is a [[empty set|nonempty]] [[Set (mathematics)|set]], ''M'' is a left ''R''-module, and ''M''^''S'' is the collection of all [[function (mathematics)|function]]s {{nowrap|''f'' : ''S'' → ''M''}}, then with addition and scalar multiplication in ''M''^''S'' defined pointwise by {{nowrap|1=(''f'' + ''g'')(''s'') = ''f''(''s'') + ''g''(''s'')}} and {{nowrap|1=(''rf'')(''s'') = ''rf''(''s'')}}, ''M''^''S'' 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'' {{nowrap|''h'' : ''M'' → ''N''}} (see below) is an ''R''-module (and in fact a ''submodule'' of ''N''^''M'').

*If ''X'' is a [[smooth manifold]], then the [[smooth function]]s from ''X'' to the [[real number]]s form a ring ''C''^∞(''X''). The set of all smooth [[vector field]]s defined on ''X'' form a module over ''C''^∞(''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''^∞(''X''), and by [[Swan's theorem]], every projective module is isomorphic to the module of sections of some vector bundle; the [[category (mathematics)|category]] of ''C''^∞(''X'')-modules and the category of vector bundles over ''X'' are [[equivalence of categories|equivalent]].

*If ''R'' is any ring and ''I'' is any [[ring ideal|left ideal]] in ''R'', then ''I'' is a left ''R''-module, and analogously right ideals in ''R'' are right ''R''-modules.

*If ''R'' is a ring, we can define the [[opposite ring]] ''R''^op, which has the same [[underlying set]] and the same addition operation, but the opposite multiplication: if {{nowrap|1=''ab'' = ''c''}} in ''R'', then {{nowrap|1=''ba'' = ''c''}} in ''R''^op. Any ''left'' ''R''-module ''M'' can then be seen to be a ''right'' module over ''R''^op, and any right module over ''R'' can be considered a left module over ''R''^op.

* [[Glossary of Lie algebras#Representation theory|Modules over a Lie algebra]] are (associative algebra) modules over its [[universal enveloping algebra]].

*If ''R'' and ''S'' are rings with a [[ring homomorphism]] {{nowrap|''φ'' : ''R'' → ''S''}}, then every ''S''-module ''M'' is an ''R''-module by defining {{nowrap|1=''rm'' = ''φ''(''r'')''m''}}. In particular, ''S'' itself is such an ''R''-module.

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Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]] of ''M''. Then ''N'' is a '''submodule''' (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product {{nowrap|''r'' ⋅ ''n''}} (or {{nowrap|''n'' ⋅ ''r''}} for a right ''R''-module) is in ''N''.

If ''X'' is any [[subset]] of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be <math display="inline">\langle X \rangle = \,\bigcap_{N\supseteq X} N</math> where ''N'' runs over the submodules of ''M'' whichthat contain ''X'', or explicitly <math display="inline">\left\{\sum_{i=1}^k r_ix_i \mid r_i \in R, x_i \in X\right\}</math>, which is important in the definition of [[tensor product of modules|tensor products of modules]].<ref>{{Cite web|url=http://people.maths.ox.ac.uk/mcgerty/Algebra%20II.pdf|title=ALGEBRA II: RINGS AND MODULES|last=Mcgerty|first=Kevin|date=2016}}</ref>

The set of submodules of a given module ''M'', together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [[Lattice (order)|lattice]] whichthat satisfies the '''[[modular lattice|modular law]]''':

Given submodules ''U'', ''N''₁, ''N''₂ of ''M'' such that {{nowrap|''N''₁ ⊂⊆ ''N''₂}}, then the following two submodules are equal: {{nowrap|1=(''N''₁ + ''U'') ∩ ''N''₂ = ''N''₁ + (''U'' ∩ ''N''₂)}}.

If ''M'' and ''N'' are left ''R''-modules, then a [[map (mathematics)|map]] {{nowrap|''f'' : ''M'' → ''N''}} is a '''[[module homomorphism|homomorphism of ''R''-modules]]''' if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'',

:<math>f(r \cdot m + s \cdot n) = r \cdot f(m) + s \cdot f(n)</math>.

This, like any [[homomorphism]] of mathematical objects, is just a mapping whichthat preserves the structure of the objects. Another name for a homomorphism of ''R''-modules is an ''R''-[[linear map]].

A [[bijective]] module homomorphism {{nowrap|''f'' : ''M'' → ''N''}} is called a module [[isomorphism]], and the two modules ''M'' and ''N'' are called '''isomorphic'''. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

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; Injective: [[Injective module]]s are defined dually to projective modules.

; 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.

; Torsionless: A module is called [[torsionless module|torsionless]] if it embeds into its [[dual module|algebraic dual]].

; 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), [https://books.google.com/books?id=KlMDjaJxZAkC&pg=PA4 p. 4], Def. 1</ref>

; Semisimple: A [[semisimple module]] is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.

; 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 whichthat are not simple (e.g. [[uniform module]]s).

; Faithful: A [[faithful module]] ''M'' is one where the action of each {{nowrap|''r'' ≠ 0}} in ''R'' on ''M'' is nontrivial (i.e. {{nowrap|''r'' ⋅ ''x'' ≠ 0}} for some ''x'' in ''M''). Equivalently, the [[annihilator (ring theory)|annihilator]] of ''M'' is the [[zero ideal]].

; 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, equivalently {{nowrap|1=''rm'' = 0}} implies {{nowrap|1=''r'' = 0}} or {{nowrap|1=''m'' = 0}}.

; Noetherian: A [[Noetherian module]] is a module whichthat 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.

; Artinian: An [[Artinian module]] is a module whichthat satisfies the [[descending chain condition]] on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

; Graded: A [[graded module]] is a module with a decomposition as a direct sum {{nowrap|1=''M'' = {{resize|140%|⨁}}_''x'' ''M''_''x''}} over a [[graded ring]] {{nowrap|1=''R'' = {{resize|140%|⨁}}_''x'' ''R''_''x''}} such that {{nowrap|''R''_''x''''M''_''y'' ⊂⊆ ''M''_''x''+''y''}} for all ''x'' and ''y''.

; Uniform: A [[uniform module]] is a module in which all pairs of nonzero submodules have nonzero intersection.

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=== Generalizations ===

A ring ''R'' corresponds to a [[preadditive category]] '''R''' with a single [[object (category theory)|object]]. With this understanding, a left ''R''-module is just a covariant [[additive functor]] from '''R''' to the [[category of abelian groups|category '''Ab''' of abelian groups]], and 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 [[sheaf (mathematics)|sheaves]] of O_''X''-modules (see [[sheaf of modules]]). These form a category O_''X''-'''Mod''', and play an important role in modern [[algebraic geometry]]. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O_''X''(''X'').