slides über Ideale

22 files changed, 609 insertions, 0 deletions

diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
index a70f73b..d6ce6b7 100644
--- a/vorlesungen/slides/3/Makefile.inc
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -13,6 +13,8 @@ chapter3 =								\
 	../slides/3/ringstruktur.tex					\
 	../slides/3/teilbarkeit.tex					\
 	../slides/3/ideal.tex						\
+	../slides/3/nichthauptideal.tex					\
+	../slides/3/nichthauptideal2.tex				\
 	../slides/3/idealverband.tex					\
 	../slides/3/maximalideal.tex					\
 	../slides/3/quotientenring.tex					\
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
index ea2718d..5287ffe 100644
--- a/vorlesungen/slides/3/chapter.tex
+++ b/vorlesungen/slides/3/chapter.tex
@@ -11,6 +11,8 @@
 \folie{3/ringstruktur.tex}
 \folie{3/teilbarkeit.tex}
 \folie{3/ideal.tex}
+\folie{3/nichthauptideal.tex}
+\folie{3/nichthauptideal2.tex}
 \folie{3/maximalideal.tex}
 \folie{3/idealverband.tex}
 \folie{3/quotientenring.tex}
diff --git a/vorlesungen/slides/3/images/Makefile b/vorlesungen/slides/3/images/Makefile
new file mode 100644
index 0000000..e338fcf
--- /dev/null
+++ b/vorlesungen/slides/3/images/Makefile
@@ -0,0 +1,55 @@
+#
+# Makefile -- build images
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \
+	hauptidealR.jpg hauptidealXR.jpg ring.jpg
+
+ring.png:	ring.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Oring.png ring.pov
+ring.jpg:	ring.png
+	convert ring.png -density 300 -units PixelsPerInch ring.jpg
+
+hauptideal.png:	hauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov
+hauptideal.jpg:	hauptideal.png
+	convert hauptideal.png -density 300 -units PixelsPerInch \
+		hauptideal.jpg
+
+hauptidealR.png:	hauptidealR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov
+hauptidealR.jpg:	hauptidealR.png
+	convert hauptidealR.png -density 300 -units PixelsPerInch \
+		hauptidealR.jpg
+
+hauptideal2.png:	hauptideal2.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov
+hauptideal2.jpg:	hauptideal2.png
+	convert hauptideal2.png -density 300 -units PixelsPerInch \
+		hauptideal2.jpg
+
+hauptidealX.png:	hauptidealX.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov
+hauptidealX.jpg:	hauptidealX.png
+	convert hauptidealX.png -density 300 -units PixelsPerInch \
+		hauptidealX.jpg
+
+hauptidealXR.png:	hauptidealXR.pov common.inc
+	povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov
+hauptidealXR.jpg:	hauptidealXR.png
+	convert hauptidealXR.png -density 300 -units PixelsPerInch \
+		hauptidealXR.jpg
+
+nichthauptideal.png:	nichthauptideal.pov common.inc
+	povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov
+nichthauptideal.jpg:	nichthauptideal.png
+	convert nichthauptideal.png -density 300 -units PixelsPerInch \
+		nichthauptideal.jpg
+
+ideal:	ideal.pov ideal.ini common.inc
+	rm -rf ideal
+	mkdir ideal
+	povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini
+
+
diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc
new file mode 100644
index 0000000..36c4e6b
--- /dev/null
+++ b/vorlesungen/slides/3/images/common.inc
@@ -0,0 +1,277 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.40;
+#declare O = <0, 0, 0>;
+#declare at = 0.10;
+
+#declare Xunten = -10;
+#declare Xoben = 10;
+#declare Yunten = -8;
+#declare Yoben = 8;
+#declare Zunten = 0;
+#declare Zoben = 20;
+
+#declare phi0 = 2 * pi * 290 / 360;
+
+camera {
+	location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10>
+	look_at <0, -2, 10>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-14, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+light_source {
+	<41, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+arrow(< -12.0,   0.0,  0  >, < 12.0,  0.0,  0.0 >, at, Gray)
+arrow(<   0.0,   0.0, -2.0>, <  0.0,  0.0, 22.0 >, at, Gray)
+arrow(<   0.0, -10.0,  0  >, <  0.0, 10.0,  0.0 >, at, Gray)
+
+#macro kasten()
+	box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> }
+#end
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau  = rgb<0.0,0.4,0.8>;
+#declare rot   = rgb<1.0,0.4,0.0>;
+
+#declare r = 0.4;
+
+#macro Zring()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Hauptideal()
+	union {
+		#declare A = Xunten;
+		#while (A <= Xoben + 0.5)
+			#declare B = Zunten;
+			#while (B <= Zoben + 0.5)
+				#declare Y = A + B;
+				#if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5))
+					sphere { <A, Y, B>, r }
+				#end
+				#declare B = B + 1;
+			#end
+			#declare A = A + 1;
+		#end
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro HauptidealR()
+	intersection {
+		kasten()
+		#declare n = vnormalize(< 1, -2, 1 >);
+		plane { n, 0.1 }
+		plane { -n, 0.1 }
+		pigment {
+			color blau
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Ideal2()
+	union {
+		#declare X = Xunten;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X, Y, Z>, r }
+					#declare Z = Z + 2;
+				#end
+				#declare Y = Y + 2;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealX()
+	union {
+		#declare Y = Yunten;
+		#while (Y <= Yoben + 0.5)
+			#declare Z = Zunten;
+			#while (Z <= Zoben + 0.5)
+				sphere { <0, Y, Z>, r }
+				#declare Z = Z + 1;
+			#end
+			#declare Y = Y + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro IdealXR()
+	intersection {
+		kasten()
+		plane { <1,0,0>, 0.1 }
+		plane { <-1,0,0>, 0.1 }
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro Nichthauptideal()
+	union {
+		#declare X = Xunten/2;
+		#while (X <= Xoben/2 + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = 0;
+				#while (Z <= Zoben + 0.5)
+					sphere { <2*X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 1;
+		#end
+		pigment {
+			color gruen
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro NichthauptidealKomplement()
+	union {
+		#declare X = Xunten + 1;
+		#while (X <= Xoben + 0.5)
+			#declare Y = Yunten;
+			#while (Y <= Yoben + 0.5)
+				#declare Z = Zunten;
+				#while (Z <= Zoben + 0.5)
+					sphere { <X,Y,Z>, r }
+					#declare Z = Z + 1;
+				#end
+				#declare Y = Y + 1;
+			#end
+			#declare X = X + 2;
+		#end
+		pigment {
+			color rot
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+
+
+
+
+
+
diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg
new file mode 100644
index 0000000..769f53c
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov
new file mode 100644
index 0000000..a934e57
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Hauptideal()
+
diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg
new file mode 100644
index 0000000..51823f3
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.jpg
diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov
new file mode 100644
index 0000000..9da5a1a
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptideal2.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Ideal2()
+
diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg
new file mode 100644
index 0000000..fae5840
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov
new file mode 100644
index 0000000..330e523
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+HauptidealR()
+
diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg
new file mode 100644
index 0000000..f9b4540
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov
new file mode 100644
index 0000000..d0045f9
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealX.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealX.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealX()
+
diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg
new file mode 100644
index 0000000..d8906c8
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.jpg
diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov
new file mode 100644
index 0000000..5daa3e6
--- /dev/null
+++ b/vorlesungen/slides/3/images/hauptidealXR.pov
@@ -0,0 +1,10 @@
+//
+// hauptidealXR.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+IdealXR()
+
diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini
new file mode 100644
index 0000000..66aa191
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.ini
@@ -0,0 +1,7 @@
+Input_File_Name=ideal.pov
+Initial_Frame=0
+Final_Frame=2500
+Initial_Clock=0
+Final_Clock=5
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov
new file mode 100644
index 0000000..88afaf7
--- /dev/null
+++ b/vorlesungen/slides/3/images/ideal.pov
@@ -0,0 +1,26 @@
+//
+// ideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = clock;
+#include "common.inc"
+
+#if (T < 1)
+Zring()
+#else
+    #if (T < 2)
+	Hauptideal()
+    #else
+	#if (T < 3)
+	    Ideal2()
+	#else
+	    #if (T < 4)
+		IdealX()
+	    #else
+		Nichthauptideal()
+		NichthauptidealKomplement()
+	    #end
+	#end
+    #end
+#end
diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg
new file mode 100644
index 0000000..55858d0
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.jpg
diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov
new file mode 100644
index 0000000..72a6330
--- /dev/null
+++ b/vorlesungen/slides/3/images/nichthauptideal.pov
@@ -0,0 +1,10 @@
+//
+// hauptideal.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Nichthauptideal()
+NichthauptidealKomplement()
diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg
new file mode 100644
index 0000000..27721b1
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.jpg
diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov
new file mode 100644
index 0000000..f854335
--- /dev/null
+++ b/vorlesungen/slides/3/images/ring.pov
@@ -0,0 +1,10 @@
+//
+// ring.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#declare T = 0;
+#include "common.inc"
+
+Zring()
+
diff --git a/vorlesungen/slides/3/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex
new file mode 100644
index 0000000..46074b9
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal.tex
@@ -0,0 +1,78 @@
+%
+% nichthauptideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}}
+\vspace{-10pt}
+\begin{align*}
+\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}];
+\foreach \x in {-6,...,6}{
+	\fill[color=red] (\x,\x) circle[radius=0.12];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Ideal mit zwei Erzeugenden}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+{\color{darkgreen}
+\langle 2,X\rangle
+}
+&=}
+\uncover<5->{
+{\color{red}2\cdot\mathbb{Z}[X]}
+}
+\uncover<6->{+}
+\uncover<4->{
+{\color{blue}X\cdot\mathbb{Z}[X]}
+}
+\end{align*}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}];
+\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}];
+\uncover<6->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,...,6}{
+			\fill[color=darkgreen] (\x,\y) circle[radius=0.20];
+		}
+	}
+}
+\uncover<5->{
+	\foreach \x in {-6,-4,...,6}{
+		\foreach \y in {-6,-4,...,6}{
+			\fill[color=red] (\x,\y) circle[radius=0.16];
+		}
+	}
+}
+\uncover<4->{
+	\foreach \y in {-6,...,6}{
+		\fill[color=blue] (0,\y) circle[radius=0.12];
+	}
+}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex
new file mode 100644
index 0000000..f362a4a
--- /dev/null
+++ b/vorlesungen/slides/3/nichthauptideal2.tex
@@ -0,0 +1,92 @@
+%
+% nichthauptideal2.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$}
+\vspace{-12pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);}
+
+\def\labels{
+	\fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2];
+	\node at (1.5,-0.1) {$1$};
+	\fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2];
+	\node at (-0.9,1.7) {$X$};
+	\fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2];
+	\node at (0.8,0.7) {$X^2$};
+}
+
+\only<-3>{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/ring.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+	\fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2];
+	\node at (1.5,-0.6) {$1$};
+	\fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2];
+	\node at (-0.4,1.7) {$X$};
+\end{scope}
+}
+
+\uncover<2->{
+\begin{scope}[xshift=-3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}};
+	\end{scope}
+	\node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<3->{
+\begin{scope}[xshift=-3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$};
+\end{scope}
+
+\begin{scope}[xshift=3.0cm,yshift=-1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}};
+	\end{scope}
+	\node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$};
+	\labels
+\end{scope}
+}
+
+\uncover<4->{
+\begin{scope}[xshift=3.0cm,yshift=1.9cm]
+	\begin{scope}
+	\c
+	\node at (0,0)
+	{\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}};
+	\end{scope}
+	\node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$};
+	\node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$};
+\end{scope}
+}
+
+\draw[color=gray!50] (-6.6,0) -- (6.4,0);
+\draw[color=gray!50] (0,-3.8) -- (0,3.8);
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup