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\begin{document}
\title{Homework $\#$7}
\author{Due 11/06/13 at start of class }
\maketitle
\subsection*{Problem 1} Find all ring homomorphisms $\mathbb{Z}\rightarrow \mathbb{Z}$.

\subsection*{Problem 2 (D$\&$F 7.3.10)} Determine which of the following are ideals of the ring $\mathbb{Z}[x]$. Justify your answers.
\begin{itemize}
\item[(a)] the set of all polynomials whose constant term is a multiple of 3.
\item[(b)] the set of all polynomials whose coefficient of $x^2$ is a multiple of 3.
\item[(c)] the set of all polynomials in which only even powers of x appear.
\item[(d)] the set of all polynomials whose constant term, coefficient of $x$, and coefficient of $x^2$ are $0$.
\item[(e)] the set of polynomials whose coefficients sum to zero
\item[(f)]  the set of polynomials $p(x)$ such that $p'(0)=0$, where $p'(x)$ is the usual first derivative of $p(x)$ with respect to $x$. 
\end{itemize}

\subsection*{Problem 3 (D$\&$F 7.3.17)}
Let $R$ and $S$ be nonzero rings with identity. Let $\phi:R\rightarrow S$ be a nonzero homomorphism of rings. 
\begin{itemize}
\item[(a)] Prove that if $\phi(1_R)\neq 1_S$ then $\phi(1_R)$ is a zero divisor in $S$. Deduce that if $S$ is an integral domain then every nonzero ring homomorphism from $R$ to $S$ sends the identity of $R$ to the identity of $S$.
\item[(b)] Prove that if $\phi(1_R)=1_S $ then $\phi(u)$ is a unit in $S$ and $\phi(u^{-1})=\phi(u)^{-1}$ for each unit $u$ of $R$.
\end{itemize}

\subsection*{Problem 4 (D$\&$F 7.3.26,28)} The \textit{characteristic} of a ring $R$ is the smallest postive integer $n$ such that $1+1+\cdots+1=0$ (n times) in $R$; if no such integer exists the characteristic of $R$ is said to be $0$. 
\begin{itemize}
\item[(a)] Determine the characteristic of the rings $\mathbb{Q},\mathbb{Z}[x],\mathbb{Z}/n\mathbb{Z}[x]$.
\item[(b)] Prove that if $p$ is a prime and $R$ is a commutative ring of characteristic $p$ then $(a+b)^p=a^p+b^p$ for all $a,b\in R$. (you may use D$\&$F 7.3.25).  
\item[(c)] Prove that an integral domain has characteristic $p$, where $p$ is either a prime or $0$.    
\end{itemize}

\subsection*{Problem 5 (D$\&$F 7.3.34)}  Let $I$ and $J$ be ideals of $R$. For the definition of $I+J$ and $IJ$, see page 247 of D\&F.
\begin{itemize}
\item[(a)] Prove that $I+J$ is the smallest ideal of $R$ containing both $I$ and $J$.
\item[(b)] Prove that $IJ$ is an ideal contained in $I\cap J$.
\item[(c)] Give an example where $IJ\neq I\cap J$.
\item[(d)] Prove that if $R$ is commutative with identity and $I+J=R$ then $IJ=I\cap J$.
\end{itemize} 

\subsection*{Problem 6 (D$\&$F 7.4.8)} 
Let $R$ be an integral domain. Prove that $(a)=(b)$ for some elements $a,b\in R$ if and only if $a=ub$ for some unit $u$ of $R$.

\subsection*{Problem 7 (D$\&$F 7.4.17)} You may want to read and work through problem 7.4.14 before considering this problem. Let $x^3-2x+1$ be an element of the polynomial ring $\mathbb{Z}[x]$ and use the bar notation to denote passage to the quotient ring $\mathbb{Z}[x]/(x^3-2x+1)$. Let $p(x)=2x^7-7x^5+4x^3-9x+1$.
\begin{itemize}
\item[(a)] Express $p(x)$ in the form $\overline{f(x)}$ for some polynomial $f(x)$ of degree $\leq 2$.
\item[(b)] Prove that $\mathbb{Z}[x]/(x^3-2x+1)$ is not an integral domain.
\end{itemize}


\subsection*{Problem 8 (D$\&$F 7.4.19)}Let $R$ be a finite commutative ring with identity. Prove that every prime ideal of $R$ is a maximal ideal.

\subsection*{Challenge Problem} Find an ideal $I$ such that $\mathbb{R}[x]/I\cong\mathbb{C}$. Under this isomorphism, what does $i$ correspond to?
\end{document}