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\def\C{\mathbb{C}}
\def\Q{\mathbb{Q}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\mfa{\mathfrak{a}}
\def\mfb{\mathfrak{b}}
\def\mfc{\mathfrak{c}}
\def\mfp{\mathfrak{p}}
\def\mfq{\mathfrak{q}}
\def\mfm{\mathfrak{m}}
\def\ev{\mathrm{ev}}
\def\Im{\mathrm{Im}}
\def\End{\mathrm{End}}
\def\Aut{\mathrm{Aut}}
\def\Int{\mathrm{Int}}
\def\Hom{\mathrm{Hom}}
\def\Iso{\mathrm{Iso}}
\def\GL{\mathrm{GL}}
\def\ker{\mathrm{ker }}
\def\Span{\mathrm{Span}}
\def\tr{\mathrm{tr}}
\def\Rings{\mathrm{Rings}}
\def\Alg{{F\text{-Alg}}}
\def\Ann{{\rm Ann}}
\newcommand{\unit}{1\!\!1}
\usepackage{tikz}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Exercises 2.3 Exercises
1. \(F\) be a field and let \(M_n(F)\) be the ring of \(n\times n\) matrices over \(F\text{.}\) Show \(\mathcal{Z}(M_n(F))\simeq F\text{.}\) 2. \(\Z/n\Z\) to \(\Z/m\Z\text{.}\) 3. \(\Aut_\Rings(\Z)\) and \(\Aut_\Rings(\Z/n\Z)\text{.}\) 4.
\(\Z/n\Z\not\simeq\Z/m\Z\) for \(m\neq n\)
\(\displaystyle \Z[X]\not\simeq\Q[X]\)
5. \(\Q\) to \(\Z\text{?}\) 6.
\begin{equation*}
\Z,\quad\Q,\quad\R,\quad\C,\quad\Q[X],\quad\Z/m\Z,\quad\Z/m\Z[X],\quad\Z\times\Z/m\Z
\end{equation*}
7. 8. \(F\) be a field, and let \(a_1,\ldots,a_n\in F\text{.}\) Show that \(\ev_{(a_1,\ldots,a_n)}\colon F[X_1,\ldots,X_n]\to F\) is an \(F\) -algebra homomorphism.
