Difference between revisions of "External direct sum module"
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==[[Characteristic property of the direct sum module]]== | ==[[Characteristic property of the direct sum module]]== | ||
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==See also== | ==See also== | ||
* [[Coproduct (category theory)]] | * [[Coproduct (category theory)]] | ||
Latest revision as of 14:05, 20 October 2016
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Review and demote as needed
- See Direct sum module for advice on using the internal or external form
Contents
Definition
Let [ilmath](R,+,*,0)[/ilmath] be a ring (with or without unity) and let [ilmath](M_\alpha)_{\alpha\in I} [/ilmath] be an arbitrary indexed family of [ilmath]R[/ilmath]-modules, the direct sum or external direct sum of the family is the following submodule of [ilmath]\prod_{\alpha\in I}M_\alpha[/ilmath] (the direct product module of the family [ilmath](M_\alpha)_{\alpha\in I} [/ilmath])[1]:
- [ilmath]\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}\}[/ilmath]
This is an instance of a categorical coproduct.
Notice that if [ilmath]\vert I\vert\in\mathbb{N} [/ilmath] then this agrees with the direct product module.
We of course the the canonical injections of a coproduct along with it, let [ilmath]\beta\in I[/ilmath] be given, then:
- [ilmath]i_\beta:M_\beta\rightarrow\bigoplus_{\alpha\in I}M_\alpha[/ilmath] by [ilmath]i_\beta:a\mapsto (0,\ldots,0,a,0,\ldots,0)[/ilmath], ie the tuple [ilmath](x_\alpha)_{\alpha\in I} [/ilmath] where [ilmath]x_\alpha=0[/ilmath] if [ilmath]\alpha\ne\beta[/ilmath] and [ilmath]x_\alpha=a[/ilmath] if [ilmath]\alpha=\beta[/ilmath]
Characteristic property of the direct sum module
- For any family of module homomorphisms, [ilmath](\varphi:M_\alpha\rightarrow M)_{\alpha\in I} [/ilmath]
- There exists a unique module homomorphism, [ilmath]\varphi:\bigoplus_{\alpha\in I}M_\alpha\rightarrow M[/ilmath], such that
- [ilmath]\forall\alpha\in I[\varphi\circ i_\alpha=\varphi_\alpha][/ilmath]
- There exists a unique module homomorphism, [ilmath]\varphi:\bigoplus_{\alpha\in I}M_\alpha\rightarrow M[/ilmath], such that
TODO: Mention commutative diagram and such
