Statement
[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]
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TODO: Caption
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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
Notes
References
- ↑ Abstract Algebra - Pierre Antoine Grillet