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\begin{document}

\begin{center}
{\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ 
	{\large Assignment 1}\\{Due Friday, January 15 at 08:35}
\end{center}

\begin{enumerate}
	\item Let $R$ be a ring.  An element $x$ of $R$ is called {\em nilpotent} if there exists
	an integer $m\ge 0$ such that $x^m=0$.  
	\begin{enumerate}
		\item Show that every nilpotent element of $R$ is either zero or a zero-divisor.
		\item Suppose that $R$ is commutative and let $x,y\in R$ be nilpotent and $r\in R$ arbitrary.  
		Prove that $x+y$ and $rx$ are nilpotent.
		\item Now suppose that $R$ is commutative with an identity and that $x\in R$ is nilpotent.
		Show that $1+x$ is a unit and deduce that the sum of a unit and a nilpotent element is a unit.
	\end{enumerate}
	
	\item Let $R$ be a commutative ring with 1 and let $f:=a_0+a_1x+\cdots + a_nx^n$ be an element of 
	the ring $R[x]$ ({\em i.e.} a polynomial in one variable over $R$).
	\begin{enumerate}
		\item Prove that $f$ is a unit in $R[x]$ if and only if $a_0$ is a unit in $R$ and $a_1,\ldots,a_n$
		are nilpotent.
		
		\item Prove that $f$ is nilpotent if and only if $a_0,\ldots,a_n$ are nilpotent.
		
		\item Prove that $f$ is a zero-divisor in $R[x]$ if and only if $f$ is nonzero and there exists $r\in R$
		with $r\neq 0$ satisfying $rf=0$.
	\end{enumerate}
	
	\item Let $n$ be a positive integer.
	\begin{enumerate}
		\item Determine the zero-divisors of the ring $\Z/n\Z$.  Prove your answer. 
		\item For a prime $p$, let $G:=\Z/p\Z$ as an abelian group (under addition of residue classes).
		Determine the zero divisors of the group-ring $\Z G$.  Hint: it may help to write $G$ multiplicatively.
	\end{enumerate}
	
	\item List all subrings of $\Z/60\Z$.  Which of these have an identity?

	\item Prove that $x\in M_n(\C)$ is nilpotent if and only if its only eigenvalue is zero.  
	Show in particular that every strictly upper-triangular
	matrix (i.e. zeroes along and below the main diagonal) is nilpotent.

%	\item Suppose $R$ is commutative ring with $a\neq 0$ and that $G$ is a nontrivial group ({\em i.e.} $|G|>1$).  
%	Prove that $RG$ has zero divisors.
	







\end{enumerate}
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