\documentclass{amsart}
\title{Problems for practice - Mid Term 3}
\author{}

\usepackage[margin=1.25in]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.} }

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\FF}[1]{\mathbb{F}_{#1}}
\newcommand{\rad}[1]{\operatorname{Rad}\left(#1 \right)}
\begin{document}
\maketitle

\begin{center}
{\bf All rings considered below are commutative and unital. All rings
homomorphisms are assumed to preserve identities.}
\end{center}

\problem{1} Let $R$ be a ring. Prove that the following are equivalent:
(1) $R$ is a field.
(2) The zero ideal is a maximal ideal.
(3) The set of ideals in $R$ is $\{ (0), (1)=R\}$.\\

\problem{2} Let $f : R\to S$ be a ring homomorphis. Prove or disprove:
\begin{enumerate}
\item If $R$ is a field, then $f$ is injective.
\item If $S$ is a field, then $f$ is injective.
\end{enumerate}

\problem{3} Prove that $\Q[x]$ is a principal ideal domain.\\

\problem{4} Let $R$ be a principal ideal domain. Prove that every non-zero
prime ideal of $R$ is maximal.\\

\problem{5} Let $R$ be a principal ideal domain. Prove that the set of
ideals in $R$ is same as $R/R^{\times}$. {\em Here $R^{\times}$ is the
group of units in $R$.}\\

\problem{6} What are the prime ideals in $\Q[x]/(x^3)$?\\

\problem{7} Let $R$ be a ring and $I, J\subset R$ be two ideals.
Prove that $IJ \subset I\cap J$. Give an example when this inclusion
is proper.\\

\problem{8} Keep the set up of Problem 7 above. Further assume that
$I$ and $J$ are coprime (that is, $I+J = R$). Prove that 
\begin{enumerate}
\item $IJ = I\cap J$.
\item The ring homomorphism $R \to R/I \times R/J$ is surjective.
\end{enumerate}

\problem{9} Let $R$ be a ring, $I, J\subset R$ two ideals and let
$P$ be a prime ideal such that $I\cap J\subset P$. Prove that either
$I\subset P$ or $J\subset P$.\\

\problem{10} Prove that every finite integral domain is a field.\\

\problem{11} Prove that every maximal ideal is prime. Give an example
of a prime ideal that is not maximal.\\

\problem{12} Prove that $\Q[x,y]$ is not a principal ideal domain.\\

\problem{13} Let $R$ be a ring. Prove that $R$ is a local ring if,
and only if $R\setminus R^{\times}$ is an ideal.\\

\problem{14} Compute the greatest common divisor $d(x)$ of the following
two elements of $\Q[x]$:
\[
a(x) = x^3 + 4x^2+x-6 \qquad\qquad b(x) = x^5-6x+5
\]
Write $d(x) = \alpha a(x) + \beta b(x)$ for $\alpha, \beta \in \Q[x]$.\\

\problem{15} Let $I\subset \Z[x]$ be a maximal ideal. Prove or disprove:
$\Z[x]/I$ is finite.\\

\problem{16} Prove that $R = \Z/36\Z$ is a principal ideal ring. Compute
each of the following in $R$:
(1) Prime ideals in $R$. (2) Maximal ideals in $R$.
(3) Units in $R$.\\

\problem{17} Let $R$ be a ring. Define the {\em characteristic} of $R$.
Let $n = \text{Characteristic}(R)$. Prove or disprove: the map $R\to R$
that sends $a\in R$ to $a^n\in R$ is a ring homomorphism.\\

\problem{18} Let $f : R\to S$ be a ring homomorphism. Recall that for
an ideal $J\subset S$, we defined
\[
f^*(J) := \{a\in R \text{ such that } f(a)\in J\}
\]
Prove, or provide a counterexample to each of the following statements.
\begin{enumerate}
\item $f^*(J)\subset R$ is an ideal.
\item If $J \subset S$ is a prime ideal, then $f^*(J)\subset R$ is
also prime.
\item If $J\subset S$ is a maximal ideal, then $f^*(J)\subset R$
is also maximal.\\
\end{enumerate}

\problem{19} Let $R\subset \Q$ be the set consisting of all rational
functions whose denominator is an odd number. Prove that $R$ is a local
ring. Describe its unique maximal ideal.\\

\problem{20} Let $R$ be a ring. For $a\in R$, define the following subset of $R$.
\[
I(a) := \{r\in R \text{ such that } ra = 0\}
\]
\begin{enumerate}
\item Prove that $I(a)\subset R$ is an ideal.
\item Prove that $I(a) \subset I(ab)$ for every $a,b\in R$.
\item Prove that $I(a) = R$ if, and only if $a=0$.\\
\end{enumerate}

\problem{21} Let $R$ be a ring and $S\subset R$ be a multiplicatively closed
set. {\em Recall that this means: $1\in S$, $0\not\in S$, and for every
$a,b\in S$, we have $ab\in S$.} 
\begin{enumerate}
\item What is the kernel of the ring homomorphism
$j : R\to S^{-1}R$? Recall that $j(r) = \frac{r}{1}$ for every $r\in R$.\\
\item Prove that every ideal in $S^{-1}R$ is of the following form:
\[
\left\{ \frac{a}{s} \text{ such that } a\in I, s\in S\right\}
\]
where $I\subset R$ is an ideal.
\end{enumerate}

\problem{22} Let $R$ be a ring. Consider the following subset (of nilpotent
elements of $R$):
\[
\mathcal{N}(R) = \{r\in R \text{ such that there exists } n\geq 1 \text{ so that }
r^n = 0\}
\]
Prove that $\mathcal{N}(R)$ is a proper ideal of $R$. Let $P\subset R$ be a prime
ideal. Prove that $\mathcal{N}(R)\subset P$.\\

\problem{23} Let $R$ be a ring. Recall that for an ideal $I\subset R$, 
we defined $\rad{I} \subset R$ as
\[
\rad{I} = \{r\in R \text{ such that } r^n\in I \text{ for some } n\geq 1\}
\]

\begin{enumerate}
\item Prove that $\mathcal{N}(R) = \rad{(0)}$.
\item Prove that $\rad{I} = \pi^*(\mathcal{N}(R/I))$ where $\pi : R\to R/I$
is the natural surjective ring homomorphism to the quotient ring.
(See problem 18 above for the notation $\pi^*$.)
\item Prove that $I$ is a proper ideal if, and only if $\rad{I}$ is a proper
ideal.
\item For any two ideals $I,J\subset R$, prove that
$\rad{I\cap J} = \rad{I}\cap \rad{J}$.
\item Let $P\subset R$ be a prime ideal. Prove that $\rad{P} = P$.
\item Let $I\subset P\subset R$, where $I$ is an ideal and $P$ is a prime
ideal. Prove that $\rad{I}\subset P$.
\item Give an example of a proper ideal $I\subset R$ such that $\rad{I} = I$
but $I$ is not a prime ideal.\\
\end{enumerate}

\end{document}
