External direct sum module

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Definition

Let $(R,+,*,0)$ be a ring (with or without unity) and let $(M_\alpha)_{\alpha\in I}$ be an arbitrary indexed family of $R$-modules, the direct sum or external direct sum of the family is the following submodule of $\prod_{\alpha\in I}M_\alpha$ (the direct product module of the family $(M_\alpha)_{\alpha\in I}$):

• $\bigoplus_{\alpha\in I}M_\alpha:=\{(x_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}M_\alpha\ \big\vert\ \ \vert\{x_\beta\in (x_\alpha)_{\alpha\in I}\ \vert\ x_\beta \ne 0\}\vert\in\mathbb{N}\}$

This is an instance of a categorical coproduct.

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

We of course the the canonical injections of a coproduct along with it, let $\beta\in I$ be given, then:

• $i_\beta:M_\beta\rightarrow\bigoplus_{\alpha\in I}M_\alpha$ by $i_\beta:a\mapsto (0,\ldots,0,a,0,\ldots,0)$, ie the tuple $(x_\alpha)_{\alpha\in I}$ where $x_\alpha=0$ if $\alpha\ne\beta$ and $x_\alpha=a$ if $\alpha=\beta$

Characteristic property of the direct sum module

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

TODO: Caption

Let $(R,+,*,0)$ be a ring (with or without unity) and let $(M_\alpha)_{\alpha\in I}$ be an arbitrary indexed family of $R$-modules and $\bigoplus_{\alpha\in I}M_\alpha$ their direct sum (external or internal). Let $M$ be another $R$-module. Then:

• For any family of module homomorphisms, $(\varphi:M_\alpha\rightarrow M)_{\alpha\in I}$
  • There exists a unique module homomorphism, $\varphi:\bigoplus_{\alpha\in I}M_\alpha\rightarrow M$, such that
    • $\forall\alpha\in I[\varphi\circ i_\alpha=\varphi_\alpha]$

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TODO: Mention commutative diagram and such

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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