Editing Ring (mathematics) (section)

== Modules ==
{{main|Module (mathematics)}}

The concept of a ''module over a ring'' generalizes the concept of a [[vector space]] (over a [[field (mathematics)|field]]) by generalizing from multiplication of vectors with elements of a field ([[scalar multiplication]]) to multiplication with elements of a ring. More precisely, given a ring {{mvar|R}}, an {{mvar|R}}-module {{mvar|M}} is an [[abelian group]] equipped with an [[operation (mathematics)|operation]] {{math|''R'' × ''M'' → ''M''}} (associating an element of {{mvar|M}} to every pair of an element of {{mvar|R}} and an element of {{mvar|M}}) that satisfies certain [[axiom#Non-logical axioms|axioms]]. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all {{math|''a''}}, {{math|''b''}} in {{mvar|R}} and all {{math|''x''}}, {{math|''y''}} in {{mvar|M}}, 
:{{mvar|M}} is an abelian group under addition.
:<math>\begin{align}
& a(x+y) = ax+ay \\
& (a+b)x = ax+bx \\
& 1x = x \\
& (ab)x = a(bx)
\end{align}</math>
When the ring is [[noncommutative ring|noncommutative]] these axioms define ''left modules''; ''right modules'' are defined similarly by writing {{mvar|xa}} instead of {{mvar|ax}}. This is not only a change of notation, as the last axiom of right modules (that is {{math|1=''x''(''ab'') = (''xa'')''b''}}) becomes {{math|1=(''ab'')''x'' = ''b''(''ax'')}}, if left multiplication (by ring elements) is used for a right module.

Basic examples of modules are ideals, including the ring itself.

Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the [[dimension (vector space)|dimension of a vector space]]). In particular, not all modules have a [[basis (linear algebra)|basis]].

The axioms of modules imply that {{math|1=(−1)''x'' = −''x''}}, where the first minus denotes the [[additive inverse]] in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.

Any ring homomorphism induces a structure of a module: if {{math|''f'' : ''R'' → ''S''}} is a ring homomorphism, then {{mvar|S}} is a left module over {{mvar|R}} by the multiplication: {{math|1=''rs'' = ''f''(''r'')''s''}}. If {{mvar|R}} is commutative or if {{math|''f''(''R'')}} is contained in the [[center of a ring|center]] of {{mvar|S}}, the ring {{mvar|S}} is called a {{mvar|R}}-[[algebra over a ring|algebra]]. In particular, every ring is an algebra over the integers.