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

\vspace{1em}

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

\begin{enumerate}
    \item Read Sections 1.2 and 1.3 of $\eh$. 
    \item Exercise 1.5. 
    \item Exercise 1.6.
    \item  Exercise 1.14.
    \item  Exercise 1.18.\\
    \end{enumerate} 
    Let $k$ be a field below. 
    \begin{enumerate}[resume]
    \item Let $\varphi:A\ra B$ be a $k$-algebra homomorphism. Show that $\kernel\varphi$ is a two-sided ideal of
        $A$ and $\image \varphi$ is a subalgebra of $B$.
    \item Prove that for any $k$-algebra $A$ and any element $a\in A$, the evaluation map $\mathrm{Eval}_a: k[t]\ra
        A, \sum \lambda_j t^j\mapsto \sum \lambda_ja^j$ is an algebra homomorphism.
    \item(Universal property of free algebras) Let $X$ be a set, let $k\ip{X}$ be the free algebra of $X$ over a field
        $k$, and let $\iota: X\ra k\ip{X}$ be the map with $\iota(x)=x$ for all $x\in X$. Prove that for any algebra
        $A$ and any function $\phi: X\ra A$, there is a unique algebra homomorphism $\bar\phi: k\ip{X}\ra A$ such
        that the following diagram commutes 
        (i.e., such
        that $\phi=\bar\phi\circ\iota$).
        \begin{center}
            \begin{tikzcd}
                X\arrow[r, "\phi"]\arrow[d,"\iota" left,hook ] & A\\
                k\ip{X}\arrow[ur,"\bar\phi" below right,dashed] &
        \end{tikzcd}
        \end{center}
    \item Let $Q$ be the \emph{two-loop quiver} with one vertex $v$ and two distinct loops $\alpha:a\ra
        a,\beta:a\ra a$ at $a$. Show that the free algebra $k\ip{X}=\ip{x,y}$ on the set $X=\{x,y\}$ with two
        elements is isomorphic to the path algebra $kQ$. (\emph{Hint}: Use the previous exercise to find a
        homomorphism, then argue that it is bijective.)
\end{enumerate}
\end{document}


