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{\large \textbf{ Math 31: Topics in Algebra \\ 
                   Summer 2019 - Problem Set 5 }}
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{\hfill Due:   Wednesday, July 31}}
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%\textbf{keywords:} \textit{homomorphism theorem, graph automorphisms} \\
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\textit{Instructions:} Write or type your answers neatly, and staple all pages before passing in your assignment.  Use complete sentences where appropriate.  Show your work; little credit will be given for solutions without work or justification. 

\vspace{1cm}

\noindent \textbf{\#1} Let $A$ be any nonempty set and let $\mathcal{P}(A)$ be its power set.  
\begin{enumerate}[a)]

\item Prove that $\mathcal{P}(A)$ is a ring, with addition defined by symmetric difference and multiplication defined by intersection. Is it commutative ring? 

\item Find all zero divisors of $\mathcal{P}(A)$.  Is it an integral domain?

\item Find all units of $\mathcal{P}(A)$. Is it a field?

\end{enumerate}

\vspace{1cm}


\noindent \textbf{\#2} (Properies of Rings) \begin{enumerate}[a)]

\item Prove that the only ideals of a field $F$ are $F$ and $\{0\}$.

\item Prove that every field is an integral domain.

\item Let $A$ be a ring.  An element $x \in A$ is \emph{idempotent} if $x^2 = x$.  Prove that if every element is idempotent, then $A$ is commutative.

\end{enumerate}

\vspace{1 cm}

\noindent \textbf{\#3} (Ideals and Subrings) \begin{enumerate}[a)]

\item Let $A$ be a commutative ring and $J$ an ideal of $A$.  Define $\Rad(J) = \{x \in A \, : \, x^n \in J \text{ for some } n \in \mathbb{Z}\}$.  Prove $\Rad(J)$ is an ideal of $A$.

\item Let $A$ be a ring and $f: A \to A$ a ring homomorphism.  Define $\Fix(f) = \{x \in A \, : \, f(x) = x\}$.  Prove $\Fix(f)$ is a subring of $A$.

\item Let $A$ be a ring and $f: A \to B$ a homomorphism.  Prove that if $A$ is an integral domain, then $f(1_A) = 1_B$ or $f(1_A) = 0_B$.  Prove also that if $f(1) = 1$, then the image of every unit in $A$ is a unit in $B$, and if $f(1) = 0$, then $f(x) = 0$ for all $x \in A$.

\end{enumerate}

\vspace{1cm} 

\noindent \textbf{\#4} (Quotient Rings) Let $A$ be a commutative ring with unity, and let $J$ be an ideal of $A$.  

\begin{enumerate}[a)]

\item Prove that $A/J$ has unity if and only if there exists an element $a \in A$ such that $ax-x \in J$ and $xa-x \in J$ for every $x \in A$.

\item Prove that $J$ is a prime ideal if and only if $A/J$ is an integral domain.

\end{enumerate}
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