# Characteristic property of the direct sum module

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Check then demote as needed

## Statement

 TODO: Caption [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]
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

## Proof

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Should be routine enough, see page 327 in Abstract Algebra - Grillet if stuck