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\centerline{\rm \bf  Math 301   Homework 5 (due on September 27, 2023)}\vskip.1in


\noindent{\Large\bf Section 3.1}
\vskip.1in
\noindent{\bf 12.}  Let   $\Z[i]$ denote the set $\{a+bi|a,\ b\in\Z\}$. Show that  $\Z[i]$ is a subring of $\C$. 


\vskip.1in

\noindent{\bf 22.} Define a new addition  $\oplus$ and  multiplication $\odot$ on $\Z$ by $$a\oplus b=a+b-1\mbox{ and } a\odot b=a+b-ab,$$
where the operation on the right-hand side of the equal signs are ordinary addition, subtraction,  and multiplication. Prove that, with the new operations  $\oplus$ and $\odot$,  $\Z$ is an integral domain.
 
\vskip.1in

\noindent{\bf 30.} The addition  table and part of the multiplication table for a four-element ring are given below. Use the distributive laws to complete the multiplication table.


$$\begin{array}{c|cccc}
+&w&x&y&z\\
\hline
w&w&x&y&z\\ 
x&x&y&z&w\\ 
y&y&z&w&x\\ 
z&z&w&x&y\end{array} \ \  \mbox{ and }  \ \ 
  \begin{array}{c|cccc}
\cdot&w&x&y&z\\
\hline
w&w&w&w&w\\ 
x&w&y&  & \\ 
y&w& &w& \\ 
z&w& &w&y\end{array}$$

\vskip.1in

\noindent{\bf 42.}  A {\bf division ring} is a (not necessarily commutative) ring $R$ with identity $ 1_R\neq 0_R$ that satisfies Axiom 11 and 12 (pages 48 and 49). Thus a field is a commutative division ring. See Exercise 43 for a noncommutative example. Suppose $R$ is a division ring and $a$, $b$ are nonzero elements of $R$.


\vskip.1in
\begin{itemize}
\item[(a)]  If $bb=b$, prove that   $ b=1_R$. [{\it Hint:} Let $v$ be a solution of $bx=1_R$ and note that $bv =b^2v$.]

\item[(b)] If $u$ is a  solution of the equation $ax=1_R$,  prove that $u$ is also a solution of the equation $ xa=1_R$. (Remember that $R$ may not be commutative.)  [{\it Hint:} Use part (a) with $b =ua$.]
\end{itemize} 


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\noindent{\Large\bf Section 3.2}
\vskip.1in
\noindent{\bf 12.}  Let $a,\ b$ be elements of a ring $R$. 

\vskip.1in
\begin{itemize}
\item[(a)]  Prove that the equation $a+x=b$ has a unique solution in $R$. (You must prove that there is a solution {\it and} that this solution is the only one.)

\item[(b)] If $R$ is a ring with identity and $a$ is a unit, prove that  the equation $ax=b$ has a unique solution in $R$.

\end{itemize} 
\noindent{\bf 18.}  Let  $a$ be a nonzero element of   a ring with identity. If the equation $ax=1_R$ has  a solution $u$ and the equation $ya = 1_R$ has  a solution $v$, prove that $u=v$.

\vskip.1in
\noindent{\bf 22.} 
\vskip.1in
\begin{itemize}
\item[(a)]  If $ab$ is a zero divisor in a ring $R$, prove that  $a$ or $b$ is a zero divisor.

\item[(b)] If $a$ or $b$ is a zero divisor in a commutative  ring $R$ and $ab\neq 0$, prove that  $ab$ is a zero divisor.
\end{itemize} 
\vskip.1in
\noindent{\bf 33.} Let $R$ be a ring with identity. If $ab$ and $a$ are units in $R$. Prove that $b$ is a unit.

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\noindent{\bf 40.} An element  $a$ of  a ring  is {\bf nilpotent} if $a^n=0_R$ for some positive integer $n$. Prove that $R$ has no nonzero nilpotent elements if and  only  if $0_R$ is the unique solution  of  the  equation $x^2=0_R$.

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