Editing Module (mathematics) (section)

=== Formal definition ===
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>

The operation · is called ''scalar multiplication''.  Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in ''R''. One may write <sub>''R''</sub>''M'' to emphasize that ''M'' is a left ''R''-module.  A '''right ''R''-module''' ''M''<sub>''R''</sub> is defined similarly in terms of an operation {{nowrap|· : ''M'' × ''R'' → ''M''}}.

The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by 
:<math> ( r s ) \cdot x = s \cdot ( r \cdot x ), </math> 
one gets a right-module, even if the scalars are written on the left. However, writing the scalars on the left for left-modules and on the right for right modules makes the manipulation of property 3 much easier.

Authors who do not require rings to be [[unital algebra|unital]] omit condition 4 in the definition above; they would call the structures defined above "unital left ''R''-modules". In this article, consistent with the [[glossary of ring theory]], all rings and modules are assumed to be unital.<ref name="DummitFoote">{{cite book | title=Abstract Algebra | publisher=John Wiley & Sons, Inc. |author1=Dummit, David S.  |author2=Foote, Richard M.  |name-list-style=amp | year=2004 | location=Hoboken, NJ | isbn=978-0-471-43334-7}}</ref>

An (''R'',''S'')-[[bimodule]] is an abelian group together with both a left scalar multiplication · by elements of ''R'' and a right scalar multiplication ∗ by elements of ''S'', making it simultaneously a left ''R''-module and a right ''S''-module, satisfying the additional condition {{nowrap|1=(''r'' · ''x'') ∗ ''s'' = ''r'' ⋅ (''x'' ∗ ''s'')}} for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.

If ''R'' is [[commutative ring|commutative]], then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules. Most often the scalars are written on the left in this case.