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\title{Math 720\\Fall 2010\\Homework 6}
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\centerline{\it Due Friday, November 12, 2010.}

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\noindent 1. (5 pt) Let $R$ be a finite ring. Show that if $R$ has an element that is not a zero divisor, then $R$ has an identity. Conclude that if $R$ is any finite ring, then every element of $R$ is either a zero divisor or a unit.


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\noindent 2. (5 pt) We say that a Boolean ring is a ring such that $x^2=x$ for all $x\in R$. Show that a Boolean ring is commutative and of characteristic $2$.

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\noindent 3. (5 pt) Show that if $R$ is commutative, then the set of nilpotent elements is an ideal (and show that this is not true in the noncommutative case).

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\noindent 4. Let $R$ be a nonzero ring such that for all $0\neq a\in R$, there is a unique $b\in R$ such that $aba=a$.
\begin{itemize}
\item[a)] (5 pt) Show that $R$ has no nontrivial zero divisors.
\item[b)] (5 pt) With the notation as above, show that $bab=b$.
\item[c)] (5 pt) Show that $R$ has an identity.
\item[d)] (5 pt) Show that $R$ is a division ring.
\end{itemize}

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\noindent 5. (5 pt) Suppose $R$ is commutative of prime characteristic $p>0$. Show that the function 

\[ 
\phi_n:R\longrightarrow R
\]

\noindent given by $\phi_n(x)=x^{p^n}$ is a ring homomorphism.



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