Module (mathematics): Difference between revisions

Revision as of 11:24, 17 March 2025

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=== Formal definition ===

Suppose that ''R'' is a [[Ring (mathematics)|ring]], and 1 is its multiplicative identity.

A '''left ''R''-module''' ''M''<ref group=note>''M'' isn't necessarily a subset of ''R'', and in general ''M'' and ''R'' don't need to have any common elements.</ref> 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>,