View source for Direct product module

{{Stub page|grade=A|msg=Be sure to mention the function construction (although indexed families are really just maps...)}}
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==Definition==
Let {{M|(R,+,*,0)}} be a ring (with out without [[ring with unity|unity]]), and let {{M|(M_\alpha)_{\alpha\in I} }} be an [[indexed family]] of left [[R-module|{{M|R}}-modules]]. We can construct a new module, denoted {{M|\prod_{\alpha\in I}M_\alpha}} that is a [[categorical product]] of the members of the family {{M|(M_\alpha)_{\alpha\in I} }}{{rAAPAG}}:
* {{M|\prod_{\alpha\in I}M_\alpha}} is the underlying set of the module (we define {{M|1=M:=\prod_{\alpha\in I}M_\alpha}} for convenience). This is a standard [[Cartesian product]]<ref group="Note">An alternate construction is that {{M|\prod_{\alpha\in I}M_\alpha}} consists of [[mappings]], {{M|f:I\rightarrow\bigcup_{\alpha\in I}M_\alpha}} where {{M|\forall\alpha\in I[f(\alpha)\in M_\alpha]}} this is just extra work as you should already be familiar with considering [[tuples]] as mappings.</ref>. The operations are:
*# ''Addition: ''<ref group="Note">the operation of the [[Abelian group]] that makes up a module</ref> {{M|+:M\times M\rightarrow M}} by {{M|+:((x_\alpha)_{\alpha\in I},(y_\alpha)_{\alpha\in I})\mapsto(x_\alpha+y_\alpha)_{\alpha\in I} }} (standard [[componentwise]] operation)
*# ''Multiplication/Action: '' {{M|\times:R\times M\rightarrow M}} given by {{M|\times:(r,(x_\alpha)_{\alpha\in I})\mapsto(rx_\alpha)_{\alpha\in I} }}, again standard componentwise definition.
: '''Claim 1: ''' this is indeed an {{M|R}}-module.<ref name="AAPAG"/>
With this definition we also get [[canonical projections of a categorical product|canonical projections]], for each {{M|\beta\in I}}<ref name="AAPAG"/>:
* {{M|\pi_\beta:M\rightarrow M_\beta}} given by {{M|\pi:(x_\alpha)_{\alpha\in I}\mapsto x_\beta}}
: '''Claim 2: ''' the canonical projections are [[module homomorphisms]]<ref name="AAPAG"/>
: '''Claim 3: ''' the {{M|R}}-module {{M|M}} is the unique {{M|R}}-module such that all the projections are module homomorphisms.<ref name="AAPAG"/>
==[[Characteristic property of the direct product module]]==
{{:Characteristic property of the direct product module/Statement}}
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==Proof of claims==
{{Requires proof|grade=B|msg=Do this section, or at least leave a guide, it should be routine and mirror the other instances of products}}
==See also==
* [[Direct sum of modules]] - instances of a {{link|co-product|category theory}}
** [[External direct sum of modules]]
** [[Internal direct sum of modules]]
* [[Characteristic property of the direct product module]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Abstract Algebra|Module Theory}}