Module (mathematics): Difference between revisions

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*If M_''n''(''R'') is the ring of {{nowrap|''n'' × ''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'' formforms 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.