Module (mathematics): Difference between revisions

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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'' 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>

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#<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 _''R''''M'' to emphasize that ''M'' is a left ''R''-module. A '''right ''R''-module''' ''M''_''R'' is defined similarly in terms of an operation {{nowrap|⋅· : ''M'' × ''R'' → ''M''}}.

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 <math> (r \cdot x) \ast s = r \cdot ( x \ast s ) </math> 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.