Difference between revisions of "Characteristic property of the direct sum module/Statement"

Latest revision as of 14:02, 20 October 2016

TODO: Caption
$\begin{xy} \xymatrix{ \bigoplus_{\alpha\in I}M_\alpha \ar[ddrr]^\varphi \ar@{<-_{)} }[dd] & & \\ & & \\ M_b \ar[rr] & & M \save (15,13)+"3,1"+{\ldots}="udots"; (8.125,6.5)+"3,1"+{M_c}="x1"; (-8.125,-6.5)+"3,1"+{M_a}="x3"; (-15,-13)+"3,1"+{\ldots}="ldots"; \ar@{<-_{)} } "x1"; "1,1"; \ar@{<-_{)} }_(0.65){i_c,\ i_b,\ i_a} "x3"; "1,1"; \ar@{<-} "x1"; "3,3"; \ar@{<-}^{\varphi_a,\ \varphi_b,\ \varphi_c} "x3"; "3,3"; \restore } \end{xy}$

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 and

$\bigoplus_{\alpha\in I}M_\alpha$ their direct sum (external or internal). Let

$M$ be another

$R$ -module. Then^[1]:

For any family of module homomorphisms, (φ:Mα→M)α∈I
- There exists a unique module homomorphism, φ:⨁α∈IMα→M, such that
  - $\forall\alpha\in I[\varphi\circ i_\alpha=\varphi_\alpha]$

TODO: Mention commutative diagram and such