% Template for the handout



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\newcommand{\MYhref}[3][blue]{\href{#2}{\color{#1}{#3}}}%

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\usepackage{amsfonts}
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\usepackage{mathtools}

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\theoremstyle{plain}
\newtheorem{thm}{Definition}[section]
\newtheorem{lemm}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition} 
\newtheorem{theo}[thm]{Theorem}
\newtheorem{Cor}[thm]{Corollary}

\theoremstyle{definition}
\newtheorem{defi}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{ex}[thm]{Example}
\newtheorem{Conc}[thm]{Conclusion}
\newtheorem{nota}[thm]{Notation}

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\def\QQ{{\mathbb Q}}
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}


\def\Name{I} % Your name
\def\Date{dd.mm.2022} % Date
\def\Title{Title} % Title of your talk
% The headers are automatically generated from this information.

\begin{document} % Begin the document

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\Name \hfill \Date\par
\vspace{0.25\baselineskip}
\hrule
\vspace{\baselineskip}
\begin{center}
{\LARGE\textbf{\Title}}\par
\vspace{0.25\baselineskip}
{\large\textsc{Seminar: Affine Algebraic Groups -- Wintersemester 2022/23}}
\end{center}
\vspace{0.125\baselineskip}
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\section{Sample Section} % Create a new section, numbering is done automatically

Here is a small example.

\begin{defi}[Prime ideal] % Creation of a definition, numbering takes place automatically, optional additional information using \begin{defi}[further description]
Let $A$ be a ring(i.e. a commutative ring with identity). A prime ideal $\mathtt{p} \subset A$, is an ideal of $A$ s.t. if $ab \in \mathtt{p}$, then necessarily $a \in \mathtt{p}$ or $b \in /mathtt{p}$. \\

\end{defi}
\begin{ex}
Let $A=\mathbb{Z}$ be the ring of integers. Then for any prime number $p \in \mathbb{Z}$, the ideal $p\mathbb{Z}$ is a prime ideal. 
\end{ex}
 
\begin{defi}[Radical]
Let $A$ be a ring. The radical of $A$, denoted $Rad(A)$, is the set of all $x \in A$ s.t. $x^{n}=0$ for some $n \in \mathbb{Z}$. 
\end{defi}
\begin{ex}
Let $A= \mathbb{Z}[X]/(X^{2})$. Then $Rad(A)$ is the ideal $(X)$ generated by the image of $X$ in $A$.
\end{ex}


\begin{lemm}
Let $A$ be a ring. \\
(a) The set $Rad(A) \subset A$ is an ideal of $A$. \\
(b) Moreover we have an equality of sets $Rad(A) = \bigcap _{\mathtt{p} \text{ prime ideal of }A} \mathtt{p}$. 
\end{lemm}

\begin{proof} % Give the proof here
Blah blah blah \dots.
\end{proof}

Example citation: \cite[Chapter 6]{Wat}.

\begin{thebibliography}{widest-label} % References are listed here

\bibitem[Wat]{Wat}
W.C. Waterhouse:
\emph{Introduction to Affine Algebraic Groups}.


\end{thebibliography}
\end{document} % Ende des Dokuments