Characteristic property of the direct product module

Stub grade: B

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Flesh out, prove... at least check before demoting

AKA: The "universal property of the direct product module".

Statement

$\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}M_\alpha \ar[dd] \\ & & \\ M \ar[uurr]^\varphi \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.65){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{\varphi_c,\ \varphi_b,\ \varphi_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}$

TODO: Description

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. Let $\prod_{\alpha\in I}M_\alpha$ be their direct product, as usual. Then:

• For any $R$-module, $M$ and
  • For any indexed family $(\varphi_\alpha:M\rightarrow M_\alpha)_{\alpha\in I}$ of module homomorphisms
    • There exists a unique morphism, $\varphi:M\rightarrow\prod_{\alpha\in I}M_\alpha$ such that:
      • $\forall\alpha\in I[\pi_\alpha\circ\varphi=\varphi_\alpha]$

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TODO: Link to diagram, this basically says it all though!

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Proof

Grade: B

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