Module (mathematics): Difference between revisions

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In [[abstract algebra]], the notion of a '''module''' is the common generalizations of two of the most important notions in algebra, [[vector space]] and [[abelian group]].

In [[abstract algebra]], a '''module''' is a generalization of a [[vector space]]. In a vector space the set of [[scalars]] forms a [[field_(mathematics)|field]], whereas in a module the scalars need only be a [[ring_(mathematics)|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 (linear algebra)|basis]].

==Motivation==

In a vector space, the set of [[scalars]] forms a [[field_(mathematics)|field]], which acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. If we ignore this operation, the vectors themselves form an ''abelian group'' under [[vector addition]].

In a module, the scalars need only be a [[ring_(mathematics)|ring]], so this concept represents a signficant generalization.

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 "nice" 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, behave significantly differently from vector spaces in some respects.

== Definition ==

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*If ''K'' is a [[field (mathematics)|field]], then the concepts "''K''-[[vector space]]" and ''K''-module are identical.

*AThe concept of a '''Z'''-module isagrees essentiallywith the same thingnotion asof 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'').

*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. Modules of this type are called [[free]] and the number ''n'' is then the [[rank]] of the free module.

*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''.