Characteristic property of the direct product module/Statement
From Maths
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[Note 1], [ilmath]\varphi:M\rightarrow\prod_{\alpha\in I}M_\alpha[/ilmath] such that:
- 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!