External direct sum module

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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

[ilmath]\begin{xy} \xymatrix{ \bigoplus_{\alpha\in I}M_\alpha \ar[ddrr]^\varphi \ar@{<-_{)} }[dd] & & \\ & & \\ M_b \ar[rr] & & M \save (15,13)+"3,1"*+{\ldots}="udots"; (8.125,6.5)+"3,1"*+{M_c}="x1"; (-8.125,-6.5)+"3,1"*+{M_a}="x3"; (-15,-13)+"3,1"*+{\ldots}="ldots"; \ar@{<-_{)} } "x1"; "1,1"; \ar@{<-_{)} }_(0.65){i_c,\ i_b,\ i_a} "x3"; "1,1"; \ar@{<-} "x1"; "3,3"; \ar@{<-}^{\varphi_a,\ \varphi_b,\ \varphi_c} "x3"; "3,3"; \restore } \end{xy}[/ilmath]

TODO: Caption

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 and [ilmath]\bigoplus_{\alpha\in I}M_\alpha[/ilmath] their direct sum (external or internal). Let [ilmath]M[/ilmath] be another [ilmath]R[/ilmath]-module. Then[1]:
  • 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]

TODO: Mention commutative diagram and such

See also



  1. 1.0 1.1 Abstract Algebra - Pierre Antoine Grillet