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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>
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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.
== Examples ==
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Suppose ''M'' is a left ''R''-module and ''N'' is a [[subgroup]] of ''M''. Then ''N'' is a '''submodule''' (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product {{nowrap|''r'' ⋅ ''n''}} (or {{nowrap|''n'' ⋅ ''r''}} for a right ''R''-module) is in ''N''.
If ''X'' is any [[subset]] of an ''R''-module ''M'', then the submodule spanned by ''X'' is defined to be <math display="inline">\langle X \rangle = \,\bigcap_{N\supseteq X} N</math> where ''N'' runs over the submodules of ''M'' that contain ''X'', or explicitly <math display="inline">\
The set of submodules of a given module ''M'', together with the two binary operations + (the module spanned by the union of the arguments) and ∩, forms a [[Lattice (order)|lattice]] that satisfies the '''[[modular lattice|modular law]]''':
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