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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-16 15:48:10 +0100 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-03-16 15:48:10 +0100 |
commit | 4614294614e6f6b38e0ca86e77871e75b4c26071 (patch) | |
tree | 23ac9079936fd3b79e790897c690146dec577eb0 /vorlesungen/slides/3/quotientenring.tex | |
parent | add new slide (diff) | |
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diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex new file mode 100644 index 0000000..4aa9e43 --- /dev/null +++ b/vorlesungen/slides/3/quotientenring.tex @@ -0,0 +1,59 @@ +% +% Quotientenring.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quotientenring} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quotientenring} +$I\subset R$ ein Ideal +\\ +\uncover<2->{ +$R/I$ hat eine Ringstruktur: +\begin{align*} +\uncover<3->{\pi(s)&=s+I} +\\ +\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\ + &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I} +\\ +\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\ + &\uncover<7->{=s+r+I=\pi(s+r)} +\end{align*}} +\end{block} +\vspace{-15pt} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Beispiele} +\begin{itemize} +\item +$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$, +$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$ +\item<8-> +$p\in\Bbbk[X]$ +$\Rightarrow$ +$\Bbbk[X]/(p)$ ist ein Ring +\end{itemize} +\end{block}} +\uncover<9->{% +\begin{block}{Algebraideal} +$I\subset A$ +\begin{itemize} +\item<10-> +$I$ ein Unterraum von $A$ als Vektorraum +\item<11-> +$I$ ein Ideal von $A$ als Ring +\end{itemize} +\end{block}} +\uncover<12->{% +\begin{block}{Quotientenalgebra} +$A/I$ ist eine Algebra +\end{block}} +\end{column} +\end{columns} +\end{frame} |