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a/sose2020/la/uebungen/la11.tex b/sose2020/la/uebungen/la11.tex new file mode 100644 index 0000000..2a1a72f --- /dev/null +++ b/sose2020/la/uebungen/la11.tex @@ -0,0 +1,299 @@ +\documentclass[uebung]{../../../lecture} + +\title{Lineare Algebra II: Übungsblatt 11} +\author{Dominik Daniel, Miriam Philipp, Christian Merten} + +\begin{document} + +\punkte[40] + +\begin{aufgabe} + Sei $R$ ein nullteilerfreier Ring und $M$ ein $R$-Modul. + \begin{enumerate}[(a)] + \item Beh.: Auf der Menge $R \times (R \setminus \{0\})$ wird durch $(r_1, s_1) \sim (r_2, s_2) \iff r_1s_2 = r_2s_1$ + eine Äquivalenzrelation definiert. + \begin{proof} + Reflexivität und Symmetrie sind klar. + + Seien weiter $r_1, r_2, r_3 \in R$ und $s_1, s_2, s_3 \in R\setminus \{0\} $ mit + $(r_1, s_1) \sim (r_2, s_2)$ und $(r_2, s_2) \sim (r_3, s_3)$. Dann gilt + \[ + r_1 s_2 = r_2 s_1 \land r_2 s_3 = s_2 r_3 + .\] + Damit folgt + \begin{salign*} + \underbrace{r_1 s_2}_{= r_2 s_1} s_3 = \underbrace{r_2 s_3}_{= s_2 r_3} s_1 + = s_2 r_1 s_3 \\ + \implies s_2(r_1 s_3) = s_2 (s_1 r_3) \\ + \implies s_2 (r_1 s_3 - s_1 r_3) = 0 + .\end{salign*} + Da $s_2 \neq 0$ und $R$ nullteilerfrei, folgt + \begin{salign*} + r_1 s_3 - s_1 r_3 = 0 \implies r_1 s_3 = s_1 r_3 \implies (r_1, s_1) \sim (r_3, s_3) + .\end{salign*} + Das zeigt die Transitivität von $\sim$ und damit die Behauptung. + \end{proof} + \item Beh.: Die Operationen + \[ + \frac{r_1}{s_1} + \frac{r_2}{s_2} \coloneqq \frac{r_1s_2 + r_2s_1}{s_1 s_2} + \quad \text{und} \quad + \frac{r_1}{s_1} \cdot \frac{r_2}{s_2} \coloneqq \frac{r_1 r_2}{s_1s_2} + \] sind wohldefiniert. + \begin{proof} + Seien $r_1, \tilde{r}_1, r_2 \in R$ und $s_1, \tilde{s}_1, s_2 \in R \setminus \{0\} $ + mit $(r_1, s_1) \sim (\tilde{r}_1, \tilde{s}_1)$. + + Z.z.: $(r_1s_2 + r_2s_1, s_1s_2) \sim (\tilde{r}_1 s_2 + r_2 \tilde{s}_1, \tilde{s}_1 s_2)$. + Es ist + \begin{salign*} + (r_1 s_2 + r_2s_1) \tilde{s}_1 s_2 &= r_1 s_2 \tilde{s}_1 s_2 + r_2 s_1 \tilde{s}_1 s_2 \\ + &\stackrel{r_1 \tilde{s}_1 = \tilde{r}_1 s_1}{=} + s_1 s_2 \tilde{r}_1 s_2 + r_2 s_1 \tilde{s}_1 s_2 \\ + &= s_1s_2(\tilde{r}_1 s_2 + r_2 \tilde{s}_1) + .\end{salign*} + Damit folgt die Behauptung. + + Z.z.: $(r_1r_2, s_1s_2) \sim (\tilde{r}_1r_2, \tilde{s}_1 s_2)$. + Es ist + \begin{salign*} + r_1 r_2 \tilde{s}_1 s_2 + &\stackrel{r_1 \tilde{s}_1 = \tilde{r}_1 s_1}{=} s_1 s_2 \tilde{r}_1 r_2 + .\end{salign*} + Damit folgt die Behauptung. + + Wohldefiniertheit folgt für zweites Argument aus Symmetriegründen. + \end{proof} + \item Beh.: Auf der Menge $M \times (R \setminus \{0\} )$ wird durch + \[ + (x_1, r_1) \sim (x_2, r_2) \iff \exists s \in R \setminus \{0\} \text{ mit } s r_1 x_2 = s r_2 x_1 + \] eine Äquivalenzrelation definiert. + \begin{proof} + Reflexivität und Symmetrie sind klar. + + Seien weiter $x_1, x_2, x_3 \in R$ und $r_1, r_2, r_3 \in R\setminus \{0\} $ mit + $(x_1, r_1) \sim (x_2, r_2)$ und $(x_2, r_2) \sim (x_3, r_3)$. Dann ex. + $s_1, s_2 \in R \setminus \{0\} $ mit $s_1 r_1 x_2 = s_1 r_2 x_1$ und + $s_2 r_2 x_3 = s_2 r_3 x_2$. + + Definiere $s \coloneqq s_1 s_2 r_2$. Es ist $s \neq 0$, da $s_1, s_2, r_2 \in R \setminus \{0\} $ + und $R$ nullteilerfrei. Dann folgt + \begin{salign*} + s_1 s_2 r_2 \cdot r_1 x_3 = s_1 r_1 \underbrace{s_2 r_2 x_3}_{= s_2 r_3 x_2} + = s_2 r_3 \underbrace{s_1 r_1 x_2}_{s_1 r_2 x_1} + = s_1 s_2 r_2 \cdot r_3 x_1 + .\end{salign*} + Das zeigt die Transitivität von $\sim$ und damit die Behauptung. + \end{proof} + \item Beh.: Mit $R = \Z$ und $M = \Z / 2 \Z$ ist die gegebene Relation nicht + transitiv. + \begin{proof} + Es ist + \[ + (\overline{1}, 1) \sim (\overline{0}, 2) \land (\overline{0}, 2) \sim (\overline{0},1) + ,\] denn $\overline{1} \cdot 2 = \overline{0} = 1 \cdot \overline{0}$ und + $\overline{0} \cdot 1 = \overline{0} = 2 \cdot \overline{0}$. Aber + $\overline{1}\cdot 1 = \overline{1} \neq \overline{0} = 1\cdot \overline{0}$. Also + ist $(\overline{1}, 1) \not\sim (\overline{0}, 1)$ + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Seien $R$ ein nullteilerfreier Ring und $M$ ein e.e. $R$-Modul. Dann sind die folgenden + Aussagen sind äquivalent: + \begin{enumerate}[(i)] + \item $M$ ist ein Torsions-$R$-Modul + \item Es gilt $\text{Ann}(M) \neq (0)$ + \end{enumerate} + \begin{proof} + (i) $\implies$ (ii): Sei $T(M) = M$. Da $M$ e.e. existiert ein endliches ES. + $\{x_1, \ldots, x_n\} \subseteq M$ von $M$. + Da $x_1, \ldots, x_n \in M = T(M)$ existieren $s_1, \ldots, s_n \in R \setminus \{0\} $ + mit + \[ + x_1 s_1 = x_2 s_2 = \ldots = x_n s_n = 0 \quad (*) + .\] Wähle $a \coloneqq s_1 \cdot \ldots \cdot s_n$. Es ist $a \neq 0$, da + $s_1, \ldots, s_n \neq 0$ und $R$ nullteilerfrei. Sei + nun $m \in M$ beliebig. Dann ex. $\alpha_1, \ldots, \alpha_n \in R$ mit + $m = \sum_{i=1}^{n} \alpha_i x_i$. Damit folgt + \[ + a m = \sum_{i=1}^{n} a \alpha_i x_i = \sum_{i=1}^{n} \alpha_i s_1 \cdot \ldots \cdot s_n x_i + \stackrel{(*)}{=} 0 + .\] Damit ist $0 \neq a \in \text{Ann}(M)$, also $\text{Ann}(M) \neq (0)$. + + (ii) $\implies$ (i): Sei $a \in \text{Ann}(M)$ mit $a \neq 0$. Dann gilt + $\forall m \in M$: $a m = 0$. Da $a \neq 0$ folgt $m \in T(M)$, also $M = T(M)$. + \end{proof} + \item Sei $R = \Z$ und $M = \oplus_{n \in \N} \Z / 2^{n} \Z$. + + Beh.: $M$ ist ein Torsions-$R$-Modul. + \begin{proof} + Sei $m \in M$ beliebig. Dann ex. $m_i \in \Z / 2^{i} \Z$ mit + $m = (m_i)_{i \in \N}$ wobei $m_i = 0$ für fast alle $i \in \N$. + Es ex. also eine Indexmenge $I \subseteq \N$ mit $\# I < \infty$ s.d. + $m_i \neq 0$ $\forall i \in I$ und $m_i = 0$ $\forall i \in \N \setminus I$. + + Definiere nun + \[ + a \coloneqq \prod_{i \in I} 2^{i} + .\] $a$ ist wohldefiniert, da $I$ endlich ist. Außerdem gilt + $a \neq 0$ da $2^{i} \neq 0$ $\forall i \in \N$ und $\Z$ nullteilerfrei. + Weiter gilt $\forall i \in I$: $a \cdot m_i = 0$, denn $\exists r_i \in \Z$ mit + $m_i = r_i + \Z / 2^{i} \Z$. Da $2^{i} \mid a$ ex. $s_i \in \Z$ mit + $a = s_i \cdot 2^{i}$. Damit folgt + \[ + a m_i = a \left( r_i + 2^{i} \Z \right) + = s_i r_i 2^{i} + 2^{i} \Z = 2^{i} \Z = \overline{0} \in \Z / 2^{i} \Z + .\] Insgesamt folgt damit $(a m)_i = 0$ $\forall i \in \N$, also + $a m = 0$. + \end{proof} + + Beh.: $\text{Ann}(M) = (0)$. + \begin{proof} + Ang. $\exists a \in \text{Ann}(M)$ mit $a \neq 0$. Dann ist $a \in \Z$ und + es ex. $k \in \N$ s.d. $2^{k} > |a|$. Damit folgt + $2^{k} \nmid a$, also $a \not\equiv 0$ $(\text{mod } 2^{k})$ $(*)$. Wähle nun + $m \coloneqq (m_1, m_2, \ldots)$ mit + \[ + m_i = \begin{cases} + 0 & i \neq 2^{k} \\ + \overline{1} & i = 2^{k} + \end{cases} + .\] Es ist $m_i = 0$ für fast alle $i \in N$ also $m \in M$, aber + \[ + a \cdot m_k = + a \cdot \overline{1} + = + a + 2^{i} \Z \stackrel{(*)}{\neq} 0 + .\] Damit folgt $a m \neq 0$, also $a \not\in \text{Ann}(M)$ $\contr$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $M$ der $\R[t]$-Modul $\R[t]/(t^2)$. + \begin{enumerate}[(a)] + \item Beh.: $T(M) = M$. + \begin{proof} + Sei $\overline{f} \in \R[t] / (t^2)$ beliebig. Dann wähle $a \coloneqq t^2 \in \R[t]$. + Es ist $a \neq 0$ und + \[ + a m = t^2 f + (t^2) = (t^2) = 0 \in \R[t] / (t^2) + .\] + \end{proof} + Beh.: $\text{Rang}(M) = 0$. + \begin{proof} + Sei $m \in M$. Dann ist $m \in T(M)$. Also ex. $s \in \R[t] \setminus \{0\} $ + mit $s m = 0$. Also ist $m$ linear abhängig. Die + max. Anzahl l.u. Elemente in $M$ ist also $0$. Damit folgt die Behauptung. + \end{proof} + \item Beh (i).: $\mathcal{B} = \{\overline{1}, \overline{t}\} $ ist Basis von $\R[t] / (t^2)$ als $\R$-Modul. + \begin{proof} + Sei $\overline{f} \in \R[t] / (t^2)$ beliebig. $\R$ ist Körper, also ist $\R[t]$ + Euklidscher Ring. Also existieren $r, q \in \R[t]$ mit $\text{deg}(r) < \text{deg}(t^2) = 2$ + und + \[ + f = q t^2 + r + .\] + Da $\text{deg}(r) < 2$ ex. $a_0, a_1 \in \R$ mit + \[ + r = a_0 + a_1 t + .\] Damit folgt + \[ + \overline{f} = f + (t^2) = q t^2 + r + (t^2) = a_0 + a_1 t + (t^2) + = a_0 \overline{1} + a_1 \overline{1} + .\] Also ist $\mathcal{B}$ ES. von $\R[t] / (t^2)$ als $\R$-Modul. + + Seien außerdem weiter $a, b \in \R$ mit + \[ + a \overline{1} + b \overline{t} = 0 \implies a + bt + (t^2) = 0 \implies + a + bt \in (t^2) + .\] Wegen $\text{deg}(a + bt) \le 1$ folgt $a = b = 0$. Also + $\mathcal{B}$ l.u. und $\mathcal{B}$ Basis von $\R[t] / (t^2)$ als $\R$- Modul. + \end{proof} + Beh.: $\R[t] / (t^2)$ ist torsionsfreier $\R$-Modul vom Rang $2$. + \begin{proof} + Wegen (i) ist $\R[t] / (t^2)$ frei mit Rang $2$ als + $\R$-Modul und damit auch torsionsfrei nach VL. + \end{proof} + \item Beh.: $\ell(M) = 2$ mit Kompositionsfaktoren $t / (t^2)$ und $\R[t] / (t^2)$. + \begin{proof} + Nach VL sind die Untermoduln von $\R[t] / (t^2)$ gerade die Untermoduln + $N$ von $\R[t]$ mit $(t^2) \subseteq N$. Diese sind gerade + die Ideale $N$ im HIR $\R[t]$ mit $(t^2) \subseteq N$, also + $\{(t^2), (t), (1)\} $. Damit sind die Untermoduln von $\R[t] / (t^2)$ gegeben als + \[ + \left\{ (t^2)/(t^2), (t) / (t^2), (1) / (t^2)\right\} + = + \left\{ 0, (t) / (t^2), \R[t] / (t^2)\right\} + .\] Als längste Filtrierung ergibt sich damit sofort + \[ + 0 \subsetneqq (t) / (t^2) \subsetneqq \R[t] / (t^2) + .\] Also folgt $\ell(M) = 2$. Die Kompositionsfaktoren ergeben sich unter + Benutzung der Isomorphiesätze der VL als: + \begin{align*} + ( (t) / (t^2) ) / 0 &\stackrel{\sim}{=} (t) / (t^2) \\ + ( (1) / (t^2)) / ( (t) / (t^2)) &\stackrel{\sim}{=} (1) / (t) + = \R[t] / (t^2) + .\end{align*} + Diese sind nach VL einfach, da die Filtrierung Länge $2 = \ell(M)$ hat. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Seien $R$ ein Ring und $M$, $N$ zwei $R$-Moduln und $\varphi\colon M \to N$ ein $R$-Mod.hom. + \begin{enumerate}[(a)] + \item Beh.: Es gilt $\ell(\text{ker } \varphi) = \ell(\text{im }\varphi) = \ell(M)$. + \begin{proof} + Definiere $\tilde{\varphi}\colon M \to \text{im }\varphi$, $m \mapsto \varphi(m)$. Dann + ist $\tilde{\varphi}$ surjektiv. Sei weiter + $\iota \colon \text{ker } \varphi \to M$ die kanonische Inklusion. Da $\iota$ injektiv + und $\text{im }\iota = \text{ker } \varphi = \text{ker } \tilde{\varphi}$ ist die kurze Folge + \[ + 0 \to \text{ker } \varphi \to M \to \text{im } \varphi \to 0 + \] exakt. Also folgt $\ell(\text{ker } \varphi) + \ell(\text{im }\varphi) = \ell(M)$. + \end{proof} + \item Beh.: Für $\ell(M) < \infty$ gilt $\ell(L) < \ell(M)$ für jeden echten $R$-Untermodul + $L \subsetneqq M$. + \begin{proof} + Sei $\ell(M) = n$ und $L \subsetneqq M$ Untermodul. Ang.: $\ell(L) \ge n$. Dann + ex. Filtrierung von $L$ der Länge $n$: + \[ + 0 \subsetneqq L_1 \subsetneqq L_2 \subsetneqq \ldots \subsetneqq L_n = L + .\] Dann ist aber + \[ + 0 \subsetneqq L_1 \subsetneqq L_2 \subsetneqq \ldots \subsetneqq L \subsetneqq M + \] eine Filtrierung von $M$ der Länge $n+1$ $\contr$. + \end{proof} + \item Beh.: Für $\ell(M) < \infty$ und $N = M$ gilt + \[ + \varphi \text{ injektiv } \iff \varphi \text{ surjektiv } \iff \varphi \text{ bijektiv} + .\] + \begin{proof} + (i) $\implies$ (ii): Sei $\varphi$ injektiv. Dann ist $\text{ker } \varphi = 0$. Also + $\ell(\text{ker } \varphi) = 0$. Damit folgt aus (a) + \[ + \ell(M) = \ell(\text{ker } \varphi) + \ell(\text{im } \varphi) = + \ell(\text{im } \varphi) + .\] Da $\ell(M) < \infty$ und $\text{im }\varphi$ Untermodul von $M$, aber + $\ell(\text{im } \varphi) = \ell(M)$ folgt mit (b), dass $\text{im }\varphi$ kein + echter Untermodul von $M$ ist. Also folgt $\text{im }\varphi = M$, also + $\varphi$ surjektiv. + + (ii) $\implies$ (iii): Sei $\varphi$ surjektiv. g.z.z. $\varphi$ injektiv. + Es folgt aus (a): + \begin{align*} + \ell(M) &= \ell(\text{im }\varphi) + \ell(\text{ker } \varphi) \\ + &= \ell(M) + \ell(\text{ker }\varphi) + .\end{align*} + Da $\ell(M) \in \N_0$ folgt $\ell(\text{ker } \varphi) = 0$. Also nach VL + $\text{ker }\varphi = 0$, also $\varphi$ injektiv. + + (iii) $\implies$ (i): trivial. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}