Editing Module (mathematics) (section)

=== Relation to representation theory ===

A representation of a group ''G'' over a field ''k'' is a module over the [[group ring]] ''k''[''G''].

If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map {{nowrap|''M'' → ''M''}} that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a [[group homomorphism|group endomorphism]] of the abelian group {{nowrap|(''M'', +)}}.  The set of all group endomorphisms of ''M'' is denoted End<sub>'''Z'''</sub>(''M'') and forms a ring under addition and [[function composition|composition]], and sending a ring element ''r'' of ''R'' to its action actually defines a [[ring homomorphism]] from ''R'' to End<sub>'''Z'''</sub>(''M'').

Such a ring homomorphism {{nowrap|''R'' → End<sub>'''Z'''</sub>(''M'')}} is called a ''representation'' of the abelian group ''M'' over the ring ''R''; 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 ''M'' over ''R''. Such a representation {{nowrap|''R'' → End<sub>'''Z'''</sub>(''M'')}} may also be called a ''ring action'' of ''R'' on ''M''.

A representation is called ''faithful'' if the map {{nowrap|''R'' → End<sub>'''Z'''</sub>(''M'')}} is [[injective]]. In terms of modules, this means that if ''r'' is an element of ''R'' such that {{nowrap|1=''rx'' = 0}} for all ''x'' in ''M'', then {{nowrap|1=''r'' = 0}}. Every abelian group is a faithful module over the [[integer]]s or over the [[Modular arithmetic|ring of integers modulo ''n'']], '''Z'''/''n'''''Z''', for some ''n''.