%%% Template for Math 4580 homework, after Eric J. Malm's hmcpset class
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%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
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\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
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\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Inn}{\operatorname{Inn}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{21}  % the homework number
\newcommand\DUEDATE{12/1/2023}  % the due date
\newcommand\COURSE{Math 4580}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
Let $S = \{ f \colon \RR \to \RR \}$ be the set of all functions from $\RR$ to itself. Addition and multiplication of functions in $S$ are defined \emph{pointwise}, so that if $f, g \in S$, then $f+g$ is the function
\[ (f+g)(x) = f(x) + g(x) \]
and $fg$ is the function
\[ (fg)(x) = f(x) g(x). \]
\begin{enumerate}
\item Prove that $S$ is a ring under these operations.
\item Does $S$ have an identity element? Is $S$ commutative?
\item Let $a \in \RR$. Define a function
\begin{align*}
\varphi_a \colon S &\to \RR \\
f &\mapsto f(a).
\end{align*}
That is, $\varphi_a$ is the \textbf{evaluation at $a$} function. Prove that $\varphi_a$ is a ring homomorphism.
\item Describe the kernel $\ker \varphi_a$.
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[2]
\begin{enumerate}
\item Describe all \emph{group} homomorphisms $\varphi \colon \ZZ \to \ZZ$.

\item Describe all \emph{ring} homomorphisms $\varphi \colon \ZZ \to \ZZ$.

[\textsc{hint:} In both cases, what can $\varphi(1)$ be?]
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[3 (Judson 16.28)]
A ring $R$ is called a \textbf{Boolean ring} if $a^2 = a$ for every $a \in R$. Prove that every Boolean ring is commutative.
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[4]
Let $\ZZ[x]$ be the ring of polynomials in the variable $x$ with integer coefficients. Determine which of the following are ideals in $\ZZ[x]$. Provide justification for your answers.
\begin{enumerate}
\item The set of all polynomials whose constant term is odd.
\item The set of all polynomials whose constant term is even.
\item The set of all polynomials whose coefficient of $x^2$ is even.
\item The set of all polynomials whose constant term, coefficient of $x$, and coefficient of $x^2$ are zero.
\item The set of all polynomials $p(x)$ such that $p'(0) = 0$, where $p'(x)$ is the usual first derivative of $p(x)$ with respect to $x$.
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[5]
Let $E$ and $F$ be fields, and let $\varphi \colon E \to F$ be a homomorphism.
\begin{enumerate}
\item Prove that if $\varphi(1) = 0$, then the image $\varphi(E)$ is equal to $\{0\}$.
\item Prove that if $\varphi(1) \neq 0$, then $\varphi(1) = 1$.
\item Prove that if $\varphi(1) \neq 0$, then $\varphi$ is injective.
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\end{document}