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+ g \colon U &\to \text{ker } \varphi \\ + u &\mapsto f(u) + .\end{align*} + $g$ ist wohldefiniert, da $\varphi \circ f = 0$, also + $\text{Bild}(f) \subseteq \text{ker } \varphi$. Da $f$ $R$-Modulhom., ist auch + $g$ $R$-Modulhom. + + Dazu sei weiter $u \in U$ beliebig. Dann ist $f(u) = g(u) = \iota(g(u))$, also + $f = \iota \circ g$. + \item Eindeutigkeit: Seien $g$, $g'\colon U \to \text{ker } \varphi$ R-Modulhom.s mit + $f = \iota \circ g = \iota \circ g'$. Dann gilt $\forall u \in U$: + $g'(u) = \iota(g'(u)) = f(u) = \iota(g(u)) = g(u)$. Also folgt + $g = g'$. + \end{enumerate} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + Seien $R$ ein Ring und $(M_i)_{i \in I}$ Familie von freien $R$-Moduln. Sei + $M = \bigoplus_{i \in I} M_i$. Sei weiter $(x_{i,j})_{j \in J_i}$ eine Basis von $M_i$ und + \[ + K \coloneqq \bigcup_{i \in I} \left( \{i\} \times J_i \right) + .\] Betrachte $(x_{i,j})_{(i,j) \in K}$ via der kanonischen Inklusionen $q_i \colon M_i \to M$ als + Familie von Elementen von $M$. Sei $N$ ein Modul mit Familie von Elementen $(y_{i,j})_{(i,j) \in K}$ + aus N. + \begin{enumerate}[(a)] + \item Beh.: $\forall i \in I$ gibt es einen eindeutigen $R$-Modulhomomorphismus $f_i \colon M_i \to N$ + mit $f_i(x_{i,j}) = y_{i,j}$ $\forall j \in J_i$. + \begin{proof} + Sei $i \in I$ beliebig. Nach Vorr. ist $M_i$ frei mit Basis $(x_{i,j})_{j \in J_i}$. + Also erfüllt nach VL $(M_i, (x_{i,j})_{j \in J_i})$ die Eigenschaft (UF). Damit folgt, + angewendet auf $N$ und die Familie $(y_{i,j})_{(i,j) \in K}$ die Behauptung. + \end{proof} + \item Beh.: $M$ ist frei. + \begin{proof} + Nach VL g.z.z., dass $(M, (x_{i,j})_{(i,j) \in K})$ die Eigenschaft (UF) erfüllt, also dass + genau ein $R$-Modulhom. $f\colon M \to N$ existiert mit + $f(x_{i,j}) = y_{i,j}$ $\forall (i,j) \in K$. + + Da $M$ direkte Summe der $M_i$ ist, erfüllt $M$ die Eigenschaft (US). Wende diese auf $N$ + mit den Homomorphismen $f_i\colon M_i \to N$ aus (a) an. Damit folgt + es ex. genau ein $f\colon M \to N$ mit $f_i = f \circ q_i$ $\forall i \in I$. Wegen + (a) gilt $\forall (i,j) \in K$ $f_i(x_{i,j}) = y_{i,j}$ $(*)$. + + Z.z.: $\forall i \in I$ $f_i = f \circ q_i$ $\iff$ $f(x_{i,j}) = y_{i,j}$ + $\forall (i,j) \in K$. + \begin{itemize} + \item ,,$\implies$``: Sei $(i,j) \in K$ beliebig. + \[ + f(x_{i,j}) = f(q_i(x_{i,j})) \; \stackrel{\text{Vorr.}}{=} \; f_i(x_{i,j}) \stackrel{(*)}{=} y_{i,j} + .\] + \item ,,$\impliedby$``: Sei $m \in M_i$ beliebig. Da $M_i$ frei mit Basis + $(x_{i,j})_{j \in J_i}$ ex. $(r_j)_{j \in J} \in R^{(J)}$ mit + $m = \sum_{j \in J_i} r_j x_{i,j}$. Damit folgt + \begin{salign*} + f_i(m) &= f_i\left(\sum_{j \in J_i} r_j x_{i,j} \right) \\ + &\stackrel{f_i \text{ } R\text{-Modulhom.}}{=} + \sum_{j \in J_i} r_j f_i(x_{i,j}) \\ + &\stackrel{(*)}{=} + \sum_{j \in J_i} r_j y_{i,j} \\ + &\stackrel{\text{Vorr.}}{=} \sum_{j \in J_i} r_j f(x_{i,j}) \\ + &= f \left( \sum_{i \in J_i} r_j x_{i,j} \right) \\ + &= f(m) + .\end{salign*} + \end{itemize} + Aufgrund der Eindeutigkeit von $f$ mit der Eigenschaft $f_i = f \circ q_i$ $\forall i \in I$ + und der gezeigten Äquivalenz, folgt die Existenz und Eindeutigkeit von $f$ mit der Eigenschaft + $f(x_{i,j}) = y_{i,j}$ $\forall (i,j) \in K$, also (UF). + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}