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\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}
