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Suppose that ''R'' is a [[Ring (mathematics)|ring]], and 1 is its multiplicative identity.
A '''left ''R''-module''' ''M'' consists of an [[abelian group]] {{nowrap|(''M'', +)}} and an operation {{nowrap|'''·''' : ''R'' × ''M'' → ''M''}} such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have
#<math> r \cdot ( x + y ) = r \cdot x + r \cdot y </math>,
#<math> ( r + s ) \cdot x = r \cdot x + s \cdot x </math>,
#<math> ( r s ) \cdot x = r \cdot ( s \cdot x ) </math>,
#<math> 1 \cdot x = x .</math>
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