Module (mathematics): Difference between revisions

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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_'''Z'''(''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_'''Z'''(''M'').

Such a ring homomorphism {{nowrap|''R'' → End_'''Z'''(''M'')}} is called a ''representation'' of the abelian group ''RM'' over the abelian groupring ''MR''; 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 ''RM'' over ''MR''. Such a representation {{nowrap|''R'' → End_'''Z'''(''M'')}} may also be called a ''ring action'' of ''R'' on ''M''.

A representation is called ''faithful'' if and only if the map {{nowrap|''R'' → End_'''Z'''(''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 somethe [[Modular arithmetic|ring of integers modulo ''n'']], '''Z'''/''n'''''Z''', for some ''n''.

=== Generalizations ===