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\begin{document}
\begin{center}
    \textsc{Math 4140/5140. Homework 3}\\
    Due Wednesday, February 2
\end{center}

\vspace{1em}

\noindent \textbf{Note:} All numbered exercises are from Erdmann--Holm ([EH]).\\


        \noindent Let $k$ be a field below.
        \vspace{0.5em}

\begin{enumerate}
    \item  Exercise 1.6.
    \item  Exercise 1.18.
    \item Exercise 1.22. (A \emph{division algebra} is an algebra where every nonzero element is invertible
    multiplicatively.)
    \item Let $A$ be a $k$-algebra and let $I$ be a two-sided ideal of $A$. Prove that the natural projection map
        $\pi: A\ra A/I$ given by $\pi(a)=a+I$ for all $a\in A$ is an algebra homomorphism.
        \item Prove the second isomorphism theorem for algebras
            (as stated in the notes of Lecture 7). You don't need to restate the theorem, and you may use the first
            isomorphism theorem without proof, but make sure you prove both             parts of the second isomorphism theorem.
        \item 
            Recall the notion of \emph{principal ideals} in an algebra from Example 1.19 in [EH]. Prove that in the
            polynomial algebra $k[x,y]$ the
            ideal $I:=\ip{x,y}$ generated by the elements $x,y$ is not a principal ideal.
\end{enumerate}
\end{document}


