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Such a ring homomorphism {{nowrap|''R'' → End<sub>'''Z'''</sub>(''M'')}} is called a ''representation'' of ''R'' over the abelian group ''M''; 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 ''R'' over it. Such a representation {{math|''R'' → End<sub>'''Z'''</sub>(''M'')}} may also be called a ''ring action'' of {{mvar|R}} on {{mvar|M}}.
A representation is called ''faithful'' if and only 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 some [[
=== Generalizations ===
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