View source for External direct sum module

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: See [[Direct sum module]] for advice on using the internal or external form
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==Definition==
Let {{M|(R,+,*,0)}} be a ring (with or without [[ring with unity|unity]]) and let {{M|(M_\alpha)_{\alpha\in I} }} be an arbitrary [[indexed family]] of [[Module|{{M|R}}-modules]], the ''direct sum'' or ''external direct sum'' of the family is the following [[submodule]] of {{M|\prod_{\alpha\in I}M_\alpha}} (the [[direct product module]] of the family {{M|(M_\alpha)_{\alpha\in I} }}){{rAAPAG}}:
* {{M|1=\bigoplus_{\alpha\in I}M_\alpha:=\{(x_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}M_\alpha\ \big\vert\ \ \vert\{x_\beta\in (x_\alpha)_{\alpha\in I}\ \vert\ x_\beta \ne 0\}\vert\in\mathbb{N}\} }}
This is an instance of a [[categorical coproduct]].

Notice that if {{M|\vert I\vert\in\mathbb{N} }} then this agrees with the [[direct product module]]. 

We of course the the [[canonical injections of a coproduct]] along with it, let {{M|\beta\in I}} be given, then:
* {{M|i_\beta:M_\beta\rightarrow\bigoplus_{\alpha\in I}M_\alpha}} by {{M|i_\beta:a\mapsto (0,\ldots,0,a,0,\ldots,0)}}, ie the [[tuple]] {{M|(x_\alpha)_{\alpha\in I} }} where {{M|1=x_\alpha=0}} if {{M|\alpha\ne\beta}} and {{M|1=x_\alpha=a}} if {{M|1=\alpha=\beta}}
==[[Characteristic property of the direct sum module]]==
{{:Characteristic property of the direct sum module/Statement}}
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==See also==
* [[Coproduct (category theory)]]
* [[Direct sum module]]
* [[Internal direct sum module]]
* [[Direct product module]]
** [[Characteristic property of the direct product module]]
==Notes==
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==References==
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{{Definition|Abstract Algebra|Module Theory}}