From 492ec1333560e8a7bfd51371df669fe8a3fdedc9 Mon Sep 17 00:00:00 2001 From: Henry Haustein Date: Sat, 8 Dec 2018 17:33:37 +0100 Subject: [PATCH] proofread bis ringe.erinnerung+wiederholung --- .../Erinnerung_und_Beispiele.tex | 101 +++++++++--------- 3. Semester/GEO/Vorlesung GEO.pdf | Bin 705071 -> 708463 bytes texmf/tex/latex/mathscript/mathscript.cls | 1 + 3 files changed, 49 insertions(+), 53 deletions(-) diff --git a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Erinnerung_und_Beispiele.tex b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Erinnerung_und_Beispiele.tex index 7e44e4b..fa9ebce 100644 --- a/3. Semester/GEO/TeX_files/Kommutative_Ringe/Erinnerung_und_Beispiele.tex +++ b/3. Semester/GEO/TeX_files/Kommutative_Ringe/Erinnerung_und_Beispiele.tex @@ -1,64 +1,65 @@ \section{Erinnerung und Beispiele} \begin{erinnerung} - Ein \begriff{Ring} ist eine abelsche Gruppe $(R,+)$ zusammen mit einer Verknüpfung $\cdot : R\times R \to R$ die Assoziativität und Distributivität erfüllt. Eine Teilmenge $\emptyset \neq S \subseteq R$ ist ein \begriff{Unterring} oder \begriff{Teilring} von $R$, wenn $S$ abgeschlossen unter Addition, Subtraktion und Multiplikation ist. Eine Abbildung $\phi : R \to R^{'}$ zwischen Ringen ist ein \begriff{Ringhomomorphismus}, wenn $\phi(r_1 + r_2) = \phi(r_1) + \phi(r_2) \text{ und } \phi(r_1 r_2) = \phi(r_1) \phi(r_2)$ und in diesem Fall ist + Ein \begriff{Ring} ist eine abelsche Gruppe $(R,+)$ zusammen mit einer Verknüpfung $\cdot : R\times R \to R$ die Assoziativität und Distributivität erfüllt. Eine Teilmenge $\emptyset \neq S \subseteq R$ ist ein \begriff{Unterring} oder \begriff{Teilring} von $R$, wenn $S$ abgeschlossen unter Addition, Subtraktion und Multiplikation ist. Eine Abbildung $\phi : R \to R'$ zwischen Ringen ist ein \begriff{Ringhomomorphismus}, wenn $\phi(r_1 + r_2) = \phi(r_1) + \phi(r_2) \text{ und } \phi(r_1 r_2) = \phi(r_1) \phi(r_2)$ und in diesem Fall ist \begin{align} - \ker(\phi)=\phi^{-1}(\{0\}) \notag + \Ker(\phi)=\phi^{-1}(\{0\}) \notag \end{align} der \begriff{Kern} von $\phi$. \end{erinnerung} \begin{remark} - In dieser Vorlesung bedeutet ``Ring'' \emph{immer} kommutativer Ring mit Einselement, d.h. $(R,\cdot)$ bildet ein kommutativer Monoid mit Einselement $1_R$. Wir fordern dann zusätzlich, dass Unterringe von $R$ das Einselement von $R$ enthalten und dass Ringhomomorphismen $\phi : R \to R^{'}$ das Einselement von $R$ auf das Einselement von $R^{'}$ abbilden. + In dieser Vorlesung bedeutet ``Ring'' \textbf{immer} kommutativer Ring mit Einselement, das heißt $(R,\cdot)$ bildet ein kommutatives Monoid mit Einselement $1_R$. Wir fordern dann zusätzlich, dass Unterringe von $R$ das Einselement von $R$ enthalten und dass Ringhomomorphismen $\phi : R \to R'$ das Einselement von $R$ auf das Einselement von $R'$ abbilden. \end{remark} \begin{example} - \begin{enumerate} + \begin{enumerate}[label=(\alph*)] \item Der Ring $\whole$ der ganzen Zahlen. \item Der Restklassenring $\whole / n \whole$ für $n \in \natur$. - \item Die Körper $\ratio, \real, \comp$. + \item Die Körper $\ratio$, $\real$, $\comp$. \item Der Nullring $R = \{0\}$ \end{enumerate} \end{example} -Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) +Seien $R$, $S$ Ringe. (Die meisten Beweise sind dem LAAG 1+2 Skript zu entnehmen!) \begin{proposition} - Ein Ringhomomorphismus $\phi: R \to S$ ist ein Isomorphismus (d.h. bijektiv), wenn es einen Ringhomomorphismus $\psi: S \to R$ mit $\psi \circ \phi = \id_R$ und $\phi \circ \psi = \id_{S}$ gibt. + Ein Ringhomomorphismus $\phi: R \to S$ ist ein Isomorphismus (das heißt bijektiv), wenn es einen Ringhomomorphismus $\psi: S \to R$ mit $\psi \circ \phi = \id_R$ und $\phi \circ \psi = \id_{S}$ gibt. \end{proposition} \begin{proposition} - Ein Ringhomomorphismus $\phi: R \to S$ ist genau dann injectiv, wenn $\ker(\phi) =\{0\}$. + Ein Ringhomomorphismus $\phi: R \to S$ ist genau dann injektiv, wenn $\Ker(\phi) =\{0\}$. \end{proposition} -\begin{definition} - Für $x \in R$ heißt \begriff{invertierbar} oder eine \begriff{Einheit}, wenn es $y\in R$ mit $xy=1$ gibt, und die $R^{\times}$ der Einheiten bildet eine Gruppe unter Multiplikation.\\ - Für $x \in R$ ist eine \begriff{Nullteiler}, wenn es $0 \neq y \in R$ mit $xy=0$ gibt, und $R$ ist \begriff{nullteilerfrei}, wenn es keinen Nullteiler $0\neq x \in R$ gibt. +\begin{definition}[invertierbar, Einheit, Nullteiler, nullteilerfrei] + Ein $x \in R$ heißt \begriff{invertierbar} oder eine \begriff{Einheit}, wenn es $y\in R$ mit $xy=1$ gibt, und die Menge $R^{\times}$ der Einheiten bildet eine Gruppe unter Multiplikation. + + Ein $x \in R$ ist ein \begriff{Nullteiler}, wenn es $0 \neq y \in R$ mit $xy=0$ gibt, und $R$ ist \begriff{nullteilerfrei}, wenn es keinen Nullteiler $0\neq x \in R$ gibt. \end{definition} \begin{example} - \begin{enumerate} + \begin{enumerate}[label=(\alph*)] \item $\whole$ ist nullteilerfrei, $\whole^{\times} = \mu_2 = \{ \pm 1 \}$. \item $\whole / n \whole$ ist genau dann nullteilerfrei, wenn $n$ prim ist. \end{enumerate} \end{example} \begin{example} - Für eine Familie von Ringen $(R_i)_{i \in I}$ wird $\prod_{i \in \Lambda} R_i$ durch komponentenweise Addition und Multiplikation zu einem Ring, genannt das \begriff{direkte Produkt} der $R_i$. Bezeichnet $1_{R_i}$ das Einselement von $R_i$, so ist $(1_{R_i})$ das Einselement von $\prod_{i \in \Lambda} R_i$ und + Für eine Familie von Ringen $(R_i)_{i \in I}$ wird $\prod_{i \in I} R_i$ durch komponentenweise Addition und Multiplikation zu einem Ring, genannt das \begriff{direkte Produkt} der $R_i$. Bezeichnet $1_{R_i}$ das Einselement von $R_i$, so ist $(1_{R_i})$ das Einselement von $\prod_{i \in I} R_i$ und \begin{align} - \left(\prod_{i\in \Lambda} R_i\right)^{\times} = \prod_{i \in I}R_i \notag + \left(\prod_{i\in I} R_i\right)^{\times} = \prod_{i \in I}R_i^\times \notag \end{align} \end{example} \begin{example} - Der \begriff{Polynomring} eine Variablen $x$ über $R$ ist + Der \begriff{Polynomring} einer Variablen $x$ über $R$ ist \begin{align} R[x] = \left\{ \sum_{i=0}^{\infty} a_i x^i \mid a_i \in R, \text{ fast alle } a_i = 0\right\} \notag \end{align} mit der Addition und Multiplikation \begin{align} \sum_{i=0}^{\infty} a_i x^i + \sum_{i=0}^{\infty} b_i x^i &= \sum_{i=0}^{\infty} (a_i + b_i) x^i \notag \\ - \left(\sum_{i=0}^{\infty} a_i x^i\right) + \left(\sum_{j=0}^{\infty} b_j x^j\right) &= \sum_{k=0}^{\infty} \left(\sum_{i+j=k}^{\infty} a_i b_j\right) x^k \notag + \left(\sum_{i=0}^{\infty} a_i x^i\right) \cdot \left(\sum_{j=0}^{\infty} b_j x^j\right) &= \sum_{k=0}^{\infty} \left(\sum_{i+j=k}^{\infty} a_i b_j\right) x^k \notag \end{align} Ist $f = \sum_{i=0}^n a_i x^i \in R[x]$ mit $a_n \neq 0$, so ist $\deg(f) = n$ der \begriff{Grad} von $f$ (mit $\deg(0) = -\infty$) und $\LC(f) = a_n$ der \begriff{Leitkoeffizient} von $f$, $f$ heißt \begriff{normiert}, wenn $\LC(f) = 1$. \end{example} @@ -66,7 +67,7 @@ Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) \begin{proposition} \proplbl{2_1_10} Seien $f,g \in R[x]$. - \begin{enumerate} + \begin{enumerate}[label=(\alph*)] \item $\deg(f + g) \leq \max\{ \deg(f), \deg(g) \}$ \item $\deg(f \cdot g) \leq \deg(f) + \deg(g)$ \item Ist $f \neq 0$ und $\LC(f)$ kein Nullteiler, so ist $\deg(fg) = \deg(f) + \deg(g)$. @@ -92,11 +93,11 @@ Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) \begin{align} 0 = \deg(1) = \deg(fg) \overset{\propref{2_1_10}}{=} \deg(f) + \deg(g) \notag \end{align} - folglich $\deg(f) = \deg(g) = 0$, d.h. $f,g \in R$. + folglich $\deg(f) = \deg(g) = 0$, das heißt $f,g \in R$. \end{itemize} \end{proof} -\begin{proposition}[Universelle Eigenschaft des Polynomrings] +\begin{proposition}[universelle Eigenschaft des Polynomrings] \proplbl{2_1_12} Ist $\phi : R \to S$ ein Ringhomomorphismus und $s \in S$, so gibt es genau einen Ringhomomorphismus $\phi_s : R[x] \to S$ mit \begin{align} @@ -105,47 +106,44 @@ Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) \end{proposition} \begin{proof} - Ist $R[x] \to S$ ein Ringhomomorphismus mit $\phi_{s}|_R = \phi$ und $\phi_{s(x)} = S$, so ist + Ist $\phi_s:R[x] \to S$ ein Ringhomomorphismus mit $\phi_{s}|_R = \phi$ und $\phi_{s}(x) = S$, so ist \begin{align} - \phi_{s}\left(\sum_{i\geq 0}^{\infty} a_i x^i\right) = \sum_{i=0}^{\infty} \phi_{s}(a_i) \phi_{s}(x^i) = \sum_{i\geq 0}^{\infty} \phi(a_i) s^i \notag + \phi_{s}\left(\sum_{i=0}^{\infty} a_i x^i\right) = \sum_{i=0}^{\infty} \phi_{s}(a_i) \phi_{s}(x^i) = \sum_{i=0}^{\infty} \phi(a_i) s^i \notag \end{align} - eindeutig bestimmt. Umgekehrt ist das so definierte $\phi_s$ ein Ringhomomorphismus (Übung), der $\phi_{s}|_R$ und $\phi_{s}(x) = s$ erfüllt. + eindeutig bestimmt. Umgekehrt ist das so definierte $\phi_s$ ein Ringhomomorphismus (Übung), der $\phi_{s}|_R$ und $\phi_{s}(x) =S$ erfüllt. \end{proof} \begin{remark} Insbesondere hat man für $a\in R$ den Einsetzungshomomorphismus: \begin{align} - \varphi_{a}: \begin{cases} + \upphi_{a}: \begin{cases} R[x] &\to R \\ f &\mapsto f(a) \end{cases}\notag \end{align} - gegeben durch $\varphi_{a}\mid_{R} = \id_R$ und $\varphi_{a}(x) = a$. Dies liefert eine Abbildung + gegeben durch $\upphi_{a}\mid_{R} = \id_R$ und $\upphi_{a}(x) = a$. Dies liefert eine Abbildung \begin{align} \begin{cases} R[x] &\to \Abb(R,R) \\ - f &\mapsto \tilde{f}, \tilde{f}(a) = \varphi_a(f) + f &\mapsto \tilde{f},\, \tilde{f}(a) = \upphi_a(f) \end{cases}\notag \end{align} - Diese Abbildung ist im Allgemeinen \emph{nicht injectiv}!!! Sei z.B. für $R = \whole / n\whole$ und $f = x^2 + x$ ist $f(0) = \overline{0}$, $f(\overline{1}) = \overline{0}$, aber $\tilde{f} = \tilde{0}$, aber $f\neq 0$. + Diese Abbildung ist im Allgemeinen \textbf{nicht injektiv}! Zum Beispiel für $R = \whole / 2\whole$ und $f = x^2 + x$ ist $f(\overline{0}) = \overline{0}$, $f(\overline{1}) = \overline{0}$, aber $\tilde{f} = \tilde{0}$, aber $f\neq 0$. \end{remark} \begin{proposition}[Polynomdivision] \proplbl{2_1_14} - Sei $0 \neq g \in R[x]$ mit $\LC(g) \in R^{\times}$. Zu jedem Polynom $f \in R[x]$ gibt es eindeutig bestimmte $q_{1}r \in R[x]$ mit $f = qg +r$ und $\deg(r) < \deg(g)$. + Sei $0 \neq g \in R[x]$ mit $\LC(g) \in R^{\times}$. Zu jedem Polynom $f \in R[x]$ gibt es eindeutig bestimmte $q,r \in R[x]$ mit $f = qg +r$ und $\deg(r) < \deg(g)$. \end{proposition} \begin{proof} - Wie im Falle $R = K$ ein Körper. + Wie im Fall $R = K$ ein Körper. \begin{itemize} - \item \textbf{Eindeutigkeit:} Sei $f = q_1 g+ r_1 = q_2 g + r_2$ und $\deg(r_1) <\deg(g) \Rightarrow r_1 - r_2 = (q_2 - q_1)g$. Da $\LC(g) \in R^{\times}$ ist $\LC(g)$ kein Nullteiler $\overset{\propref{2_1_10}}{\Rightarrow} \deg(r_1 - r_2) < \deg(g) = \deg(q_2 - q_1) + \deg(g)$ \\ + \item \textbf{Eindeutigkeit:} Sei $f = q_1 g+ r_1 = q_2 g + r_2$ und $\deg(r_1) <\deg(g) \Rightarrow r_1 - r_2 = (q_2 - q_1)g$. Da $\LC(g) \in R^{\times}$ ist $\LC(g)$ kein Nullteiler $\overset{\propref{2_1_10}}{\Rightarrow} \underbrace{\deg(r_1 - r_2)}_{<\deg(g)} = \deg(q_2 - q_1) + \deg(g)$ \\ $\Rightarrow \deg(q_2 - q_1) < 0 \Rightarrow q_1 = q_2$ und $r_1 = r_2$ - \item \textbf{Existenz:} Sei $f = \sum_{i=0}^{n} a_i x^i$, $a_n \neq 0$ und $g = \sum_{j=0}^{m} b_j x^j$ mit $b_m \neq 0$. Nach Vorraussetzung ist $b_m \in R^{\times}$ es existiert also $b_m^{-1} \in R$.\\ - Induktion nach $\deg(f) = n$: - \begin{itemize} - \item \textbf{$n < m$:} $q = 0$, $r = f$ - \item \textbf{$n\geq m$:} $f_i = f - a_n b_m^{-1} x^{n-m} \cdot g \Rightarrow \deg(f_1) < \deg(f)$ mit Induktionshypothese folgt $f_1 = q_1 \cdot g + r_1$ mit $\deg(r) < m$ $\Rightarrow f = (q_1 + a_n b_m^{-1} x^{n-m})g + r$ - \end{itemize} + \item \textbf{Existenz:} Sei $f = \sum_{i=0}^{n} a_i x^i$, $a_n \neq 0$ und $g = \sum_{j=0}^{m} b_j x^j$ mit $b_m \neq 0$. Nach Voraussetzung ist $b_m \in R^{\times}$, es existiert also $b_m^{-1} \in R$. Induktion nach $\deg(f) = n$: \\ + \emph{$n < m$:} $q = 0$, $r = f$ \\ + \emph{$n\geq m$:} $f_i = f - a_n b_m^{-1} x^{n-m} \cdot g \Rightarrow \deg(f_1) < \deg(f)$ mit Induktionshypothese folgt $f_1 = q_1 \cdot g + r_1$ mit $\deg(r) < m$ $\Rightarrow f = (q_1 + a_n b_m^{-1} x^{n-m})g + r$ \end{itemize} \end{proof} @@ -157,7 +155,7 @@ Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) \end{conclusion} \begin{proof} - Sei $f = q(x-a) + r$, $\deg(r) < \deg(x-a)$, d.h. $\deg(1) \leq 0 \Rightarrow 0 = f(a) = q(a-a) + r(a) \Rightarrow r(a) = 0$. + Sei $f = q(x-a) + r$, $\deg(r) < \deg(x-a)$, das heißt $\deg(r) \leq 0 \Rightarrow 0 = f(a) = q(a-a) + r(a) \Rightarrow r(a) = 0$. \end{proof} \begin{conclusion} @@ -166,45 +164,42 @@ Seien $R, S$ Ringe. (Meisten Beweise sind LAAG1+2 Skript zu entnehmen!) \end{conclusion} \begin{definition}[Polynomring in kommutieren Variablen] - Für eine Menge $I$ definieren wir den Monoid + Für eine Menge $I$ definieren wir das Monoid \begin{align} - \natur_{0}^{(I)} : = \left\{ (\mu_i)_{i \in I} \in \prod_{i \in I} \natur_{0} \colon \mu_i = 0 \text{ für fast alle } i \right\}\notag + \natur_{0}^{(I)} : = \left\{ (\mu_i)_{i \in I} \in \prod_{i \in I} \natur_{0} \Bigg| \mu_i = 0 \text{ für fast alle } i \right\}\notag \end{align} mit Addition \begin{align} - (\mu_i)_{i \in I} + (\nu_i)_{i \in I} :=(\mu_i+\nu_i)_{i \in I}, \notag + (\mu_i)_{i \in I} + (\nu_i)_{i \in I} :=(\mu_i+\nu_i)_{i \in I} \notag \end{align} sowie den Ring \begin{align} - R[x_i \colon i \in I] = \{ (_{\mu})_{\mu \in \natur_{0}} \colon a_{\mu} \in R, \text{ fast alle gleich } 0\} \notag + R[x_i \mid i \in I] = \{ (a_{\mu})_{\mu \in \natur_{0}^{(I)}} \mid a_{\mu} \in R, \text{ fast alle gleich } 0\} \notag \end{align} mit Addition und Multiplikation \begin{align} - (a_{\mu})_{\mu \in \natur_{0}}^{(I)} + (b_{\mu})_{\mu \in \natur_{0}}^{(I)} &:= (a_{\mu} + b_{\mu})_{\mu \in \natur_{0}}^{(I)}\notag \\ - (a_{\lambda})_{\lambda \in \natur_{0}^{(I)}}\cdot (b_{\nu})_{\nu \in \natur_{0}^{(I)}} &:= \left( \sum_{\lambda + \nu = \mu} a_{\lambda}b_{\mu}\right)_{\mu \in \natur_{0}}^{(I)}, \notag + (a_{\mu})_{\mu \in \natur_{0}^{(I)}} + (b_{\mu})_{\mu \in \natur_{0}^{(I)}} &:= (a_{\mu} + b_{\mu})_{\mu \in \natur_{0}^{(I)}}\notag \\ + (a_{\lambda})_{\lambda \in \natur_{0}^{(I)}}\cdot (b_{\nu})_{\nu \in \natur_{0}^{(I)}} &:= \left( \sum_{\lambda + \nu = \mu} a_{\lambda}b_{\mu}\right)_{\mu \in \natur_{0}^{(I)}} \notag \end{align} - gennant \begriff{Polynomring in kommutierenden Variablen} $x_i, i \in I$. Wir identifizieren den Ring $R$ mit den Unterring + genannt \begriff{Polynomring in den kommutierenden Variablen} $x_i$ mit $i \in I$. Wir identifizieren dem Ring $R$ mit den Unterring \begin{align} - \{ (r\delta_{\mu,\underline{0}})_{\mu \in \natur_{0}}^{(I)}\colon r \in R \}. \notag + \left\{ (r\delta_{\mu,\underline{0}})_{\mu \in \natur_{0}^{(I)}}\mid r \in R \right\} \notag \end{align} - Wir schreiben $x_i := (\delta_{\mu\nu})_{\mu \in \natur_{0}}^{(I)}$, $\mu := (\delta_{ij})_{i \in I}$ und $x^{\mu} := \prod_{i \in I}x_i^{\mu_i}$. Damit ist dann + Wir schreiben $x_i := (\delta_{\mu\nu})_{\mu \in \natur_{0}^{(I)}}$, $\nu := (\delta_{ij})_{j \in I}$ und $x^{\mu} := \prod_{i \in I}x_i^{\mu_i}$. Damit ist dann \begin{align} - (a_{\mu})_{\mu \in \natur_{0}}^{(I)} = \sum_{\mu \in \natur_{0}^{(I)}} a_{\mu} x^{\mu}. \notag - \end{align} - Weiter schreiben wir - \begin{align} - R[x_1, \dots, x_n] := R[x_i \colon i \in \{ i, \dots, n \}]. \notag + (a_{\mu})_{\mu \in \natur_{0}^{(I)}} = \sum_{\mu \in \natur_{0}^{(I)}} a_{\mu} x^{\mu}. \notag \end{align} + Weiter schreiben wir $R[x_1, \dots, x_n] := R[x_i \mid i \in \{ i, \dots, n \}]$. \end{definition} \begin{example} Sei $R = \whole$ und $I = \{1,2\}$, dann \begin{align} - \left( x_1 x_2 + x_2^2 \right)^2 = a_{(2,1)}x_1^2 x_2^2 + a_{(1,3)}x_1 x_2^2 + a_{(0,4)}x_2^4\notag + \left( x_1 x_2 + x_2^2 \right)^2 = a_{(2,2)}x_1^2 x_2^2 + a_{(1,3)}x_1 x_2^3 + a_{(0,4)}x_2^4\notag \end{align} - mit $a_{(2,1)} = 1$, $a_{(1,3)} = 2$ und $a_{(0,4)} = 1$ + mit $a_{(2,2)} = 1$, $a_{(1,3)} = 2$ und $a_{(0,4)} = 1$ \end{example} \begin{remark} - \propref{2_1_10} und \propref{2_1_12} kann man allgemein für $R[x_i \colon i \in I]$ anstatt $R[x]$ formulieren. Für \propref{2_1_14} - \propref{2_1_16} gibt es keine Verallgeminerung. So hat z.B. $f = x_1 - x_2$ unendlich viele Nullstellen, da $f(a,a) = 0$ für alle $a \in \whole$. + \propref{2_1_10} und \propref{2_1_12} kann man allgemein für $R[x_i \mid i \in I]$ anstatt $R[x]$ formulieren. Für \propref{2_1_14} bis \propref{2_1_16} gibt es keine Verallgemeinerung. So hat zum Beispiel $f = x_1 - x_2$ unendlich viele Nullstellen, da $f(a,a) = 0$ für alle $a \in \whole$. \end{remark} \ No newline at end of file diff --git a/3. Semester/GEO/Vorlesung GEO.pdf b/3. 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