%%This is a standard LaTeX2e article document template. personal version 12/5/200%% \documentclass[11pt,twoside]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%packages%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{empty} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amstext} \usepackage{multicol} \usepackage{amsxtra} \usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref} \hypersetup{ pdftitle = {Math 371 Assignment 2}, pdfauthor = {\textcopyright\ Bryden Cais}, pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq}, colorlinks = {true} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%formatting%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\topmargin}{0in} %%% This sets all the spacing stuff to use the page more \setlength{\oddsidemargin}{0in} %%% efficiently than the normal "article" setup would. \setlength{\evensidemargin}{0in} %%% It's OK to play with these some. \setlength{\textheight}{7.5in} %%% \setlength{\textwidth}{6.5in} %%% \setlength{\headsep}{0in} %%% \setlength{\headheight}{0in} %%% %\setlength{\footskip}{0in} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\GL}{GL} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\C}{\mathbf{C}} \begin{document} \begin{center} {\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ {\large Assignment 2}\\{Due Monday, January 25 at 08:35} \end{center} \begin{enumerate} \item Let $R$ be a ring. \begin{enumerate} \item Let $I$ be an ideal of $R$ and denote by $\pi: R\rightarrow R/I$ the natural ring homomorphism defined by $\pi(x):=x \bmod I $ ($= x + I$ using coset notation). Show that an arbitrary ring homomorphism $\phi:R\rightarrow S$ can be factored as $\phi = \psi\circ \pi$ for some ring homomorphism $\psi: R/I\rightarrow S$ if and only if $I\subseteq \ker(\phi)$, in which case $\psi$ is unique. \item Suppose that $R$ is commutative with $1$. An {\em $R$-algebra} is a ring $S$ with identity equipped with a ring homomorphism $\phi:R\rightarrow S$ mapping $1_R$ to $1_S$ such that $\mathrm{im}(\phi)$ is contained in the center of $S$ ({\em i.e.} the set $$c(S):=\{z\in S\ |\ zs=sz\ \text{for all}\ s\in S\}$$ of all elements of $S$ that commute with every other element). If $(S,\phi)$ and $(S',\phi')$ are two $R$-algebras then a ring homomorphism $f:S\rightarrow S'$ is called a {\em homomorphism of $R$-algebras} if $f(1_S)=1_{S'}$ and $f\circ \phi = \phi'$. For an $R$-algebra $(S,\phi)$ we will frequently simply write $rx$ for $\phi(r)x$ whenever $r\in R$ and $x\in S$. Prove that the polynomial ring $R[X]$ in one variable is naturally an $R$-algebra, and that if $S$ is an $R$-algebra then for any $s\in S$ there exists a unique $R$-algebra homomorphism $f:R[X]\rightarrow S$ such that $f(X)=s$. In other words, mapping $R[X]$ to $S$ is the ``same" as choosing an element $s$ of $S$. \end{enumerate} \item Let $R$ be a ring with $1$. \begin{enumerate} \item Prove that there is a unique map of rings $f_R:\Z\rightarrow R$. Conclude that every ring with $1$ is a $\Z$-algebra in a unique way. \label{first} \item For a ring $R$ with $1$, the kernel of the ring homomorphism $f_R$ as in (\ref{first}) is an ideal of $\Z$ so it has the form $c(R)\Z$ for a unique $c(R)\in \Z$ satisfying $c(R)\ge 0$. By definition, the {\em characteristic of $R$} is this integer $c(R)$. Convince yourself that when $c(R)>0$, this number is the least number of times we have to add $1\in R$ to itself to get $0\in R$. Now prove that if $R$ is a ring with $1$ that is an integral domain, then the characteristic of $R$ is either $0$ or a prime number. \item Prove that for $g:R\rightarrow S$ a homomorphism of rings with $1$ taking $1_R$ to $1_S$ the characteristic of $S$ divides the characteristic of $R$. \item Let $g:R\rightarrow S$ be a homomorphism of rings with $1$ taking $1_R$ to $1_S$. If $g$ is injective, prove that $c(R)=c(S)$. Give an example with $g$ not injective where $c(R)\neq c(S)$. \end{enumerate} \item Let $I$ and $J$ be ideals of a ring $R$. We define \begin{enumerate} \item $I+J:=\{a+b\ |\ a\in I,\ b\in J\}$ \item $IJ:=\{a_1b_1 + \cdots + a_sb_s\ |\ a\in I,\ b\in J\}$ \end{enumerate} Prove that $I+J$ is the smallest ideal of $R$ containing $I$ and $J$ and that $IJ$ is an ideal contained in the intersection $I\cap J$. Convince yourself that $I\cap J$ is an ideal of $R$, and show that if $R$ is commutative and $I+J=R$ then $IJ=I\cap J$. Show by giving examples that $IJ\neq I\cap J$ in general, and that $I\cup J$ (set-theoretic union) need not be an ideal. \item Let $R$ be a commutative ring and $I,J$ ideals of $R$. If $P$ is a prime ideal of $R$ containing $IJ$, prove that $P$ contains $I$ or $P$ contains $J$. \item Let $R$ be a commutative ring. \begin{enumerate} \item Show that the set of all nilpotent elements of $R$ ( called the {\em nilradical of $R$}) is an ideal. Hint: this is basically 1(b) from assignment 1, but be careful about showing that this set is really an abelian group under addition. \item Prove that the nilradical of $R$ is contained in the intersection of all prime ideals of $R$. \item Let $G:=\Z/p\Z$ as a group under addition (it is cyclic of order $p$). Let $\mathbf{F}_p:=\Z/p\Z$ as a ring, and note that this is a field with $p$ elements. Let $R$ be the group ring $R:=\mathbf{F}_pG$. What is the nilradical of $R$? \end{enumerate} \item Let $R$ be a commutative ring. Prove that the set of prime ideals in $R$ has minimal elements with respect to inclusion. Such minimal elements are called {\em minimal primes}. \item Let $R$ be a finite (as a set) commutative ring with 1. Prove that every prime ideal of $R$ is maximal. \item Let $\varphi:R\rightarrow S$ be a homomorphism of commutative rings and $I$ an ideal of $S$. Prove that $\varphi^{-1}(I)$ (set-theoretic inverse image) is an ideal of $R$ that is prime whenever $I$ is a prime ideal of $S$. Show that this holds with ``prime" replaced by ``maximal" provided we assume that $\varphi$ is surjective. Give a counterexample to this if we drop the surjectivity requirement. \end{enumerate} \end{document}