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*If ''K'' is a [[field (mathematics)|field]], then the concepts "''K''-[[vector space]]" and ''K''-module are identical.
*A '''Z'''-module is essentially the same thing as 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''<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. 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''<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''.
*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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