# Characteristic property of the direct product module

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**Stub grade: B**

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

- AKA: The "
*universal property of the direct product module*"^{[1]}.

## Contents

## Statement

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. Let [ilmath]\prod_{\alpha\in I}M_\alpha[/ilmath] be their direct product, as usual. Then^{[1]}:

- For any [ilmath]R[/ilmath]-module, [ilmath]M[/ilmath] and
- For any indexed family [ilmath](\varphi_\alpha:M\rightarrow M_\alpha)_{\alpha\in I} [/ilmath] of module homomorphisms
- There exists a unique morphism
^{[Note 1]}, [ilmath]\varphi:M\rightarrow\prod_{\alpha\in I}M_\alpha[/ilmath] such that:- [ilmath]\forall\alpha\in I[\pi_\alpha\circ\varphi=\varphi_\alpha][/ilmath]

- There exists a unique morphism

- For any indexed family [ilmath](\varphi_\alpha:M\rightarrow M_\alpha)_{\alpha\in I} [/ilmath] of module homomorphisms

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

## Proof

Grade: B

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The message provided is:

The message provided is:

Routine work, just gotta be bothered to do it!

## Notes

## References

- ↑
^{1.0}^{1.1}Abstract Algebra - Pierre Antoine Grillet