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\begin{document} 
\noindent
{\bf Math 307 Abstract Algebra   \quad Homework 9}

\bigskip
Six points for each question unless specified otherwise.

\begin{enumerate}

\item (a) Give an example of a subset of a ring that is a subgroup under addition but 
not a subring.

(b) Give an example of a finite non-commutative ring. 

\item  Let $R$ be a ring.

(a) Let $a \in R$ and $\langle a \rangle = \{ma: m \in \IZ\}$ be the 
additive subgroup generated by  $a$. (Here,  

Show that if $m,n$ are integers and $a,b$ are elements in a ring. Then 
$(ma)(nb) = (mn)(ab)$. [Note that for positive $m$, $ma = a+ \cdots + a$ ($m$ times);
for $(-m)a = m(-a) = (-a) + \cdots + (-a)$.]

[Hint: Discuss the cases (1) $m,n > 0$, (2) $mn = 0$,
(3) $mn< 0$, (4) $m,n < 0$.]

\item 
Let $R$ be a ring.

(a) Suppose $a \in R$. Show that $S = \{x \in R: ax=xa\}$ is a subring.

(b) Show that the center of $R$ defined by
$Z(R) = \{x \in R: ax = xa \hbox{ for all } a \in R\}$ is a subring.

\item 
Let $R$ be a ring. 

(a) Prove that $R$ is commutative if and only if $a^2-b^2 = (a+b)(a-b)$
for all $a, b \in R$.

(b) Prove that $R$ is commutative if $a^2 = a$ for all $a \in R$. 

(Such a ring is called
a Boolean ring.)

[Hint: In (b), think about $a = (x+y), (x-y)$, etc.

(c) (Extra 3 points) Give an example of a Boolean ring with 4 elements. Give an example of 
a Boolean ring with infinitely many elements.

[Hint: Consider $\IZ_2 \oplus \IZ_2$, and extend the idea.]

\item Show that every nonzero element $[k]$ of $\Z_n$ is a unit (element with multiplicative inverse) or a zero-divisor depending on gcd$(k,n)$ is 1 or larger.


\item 
(a) Show that $2+i$ in $\Z_5[i]$ has no
multiplicative inverse.

(b) Show that every nonzero
element in $\Z_7[i] = \{a + bi: a, b \in \Z_7\}$
 has a multiplicative inverse.
 
 [Hint: Show that for any nonzero $a+ib$, 
  $(a^2+b^2)^{-1}$ always exists, and $x+iy = (a^2+b^2)^{-1}(a-bi)$
  is the inverse of $a+ib$.]

(c) (Extra 4 points.) Determine those prime $p$ such that 
every nonzero element in $\Z_p[i]$
has a multiplicative inverse.


\item (a) 
Given an example of a commutative ring without zero-divisors that is not an integral domain.

(b) Find two elements $a$ and $b$ in a ring such that $a, b$ are zero-divisors, $a+b$ is
a unit.


\item  (a) Give an example to show that the characteristic of a subring 
of a ring $R$ may be different from that of  $R$.

(b) Show that the characteristic of a subdomain of an integral
domain $D$ is the same as that of $D$.

[Hint: In both parts, focus on the unity in the ring if it exists.]


\end{enumerate}

\end{document}