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\large
\centerline{\rm \bf  Math 301   Homework 4 (due on September 20, 2023)}\vskip.1in

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\noindent{\Large\bf Section 2.2}
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\noindent{\bf 6.} Solve the equation $x^2\oplus [8]\odot x = [0]$ in $\Z_9$.
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\noindent{\bf 8.} Solve the equation $x^3\oplus x^2 = [2]$ in $\Z_{10}$.
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\noindent{\bf 12.} Prove or disprove: If $[a]\odot [b]=[0]$ in $\Z_n$, then $[a]=[0]$ or $[b]=[0]$.
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\noindent{\bf 14(c).} If $p$ is a positive prime, show that the only solutions of $x^2\oplus x=[0]$ in $\Z_p$ are $[0]$ and $[p-1]$.
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\noindent{\bf 16.} Find all $[a]$ in $\Z_n$ for which the equation $[a]\odot x=[1]$ has a solution, in the case when 
\[\text{(a)}\; n=5,\qquad\text{(b)}\; n=4,\qquad \text{(c)}\; n=3,\qquad\text{(d)}\; n=6\]

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\noindent{\Large\bf Section 2.3}
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\noindent{\bf 2.} Find all zero-divisors in
\[\text{(a)}\; \Z_7,\qquad\text{(b)}\; \Z_8,\qquad \text{(c)}\; \Z_9,\qquad\text{(d)}\; \Z_{10}\]


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\noindent{\bf 6.} If $n$ is composite, prove that there is at least one zero-divisor in $\Z_n$. (See Exercise 2.)
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\noindent{\bf 10.} Prove that every nonzero element of $\Z_n$ is either a unit or a zero-divisor, but not both. [Hint: Exercise 9 provides the proof of "not both".]

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\noindent{\Large\bf Section 3.1}
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\noindent{\bf 2.} Let $R=\{0,e,b,c\}$ with addition and multiplication defined by the tables below. Assume associativity and distributivity and show that $R$ is a ring with identity. Is $R$ commutative? Is $R$ a field?
\begin{center}
\begin{tabular}{c|cccc}
+ & 0 & e & b & c \\
\hline
0 & 0 & e & b & c \\
e & e & 0 & c & b \\
b & b & c & 0 & e \\
c & c & b & e & 0
\end{tabular}
\hspace{1in}
\begin{tabular}{c|cccc}
$\cdot$ & 0 & e & b & c \\
\hline
0 & 0 & 0 & 0 & 0 \\
e & 0 & e & b & c \\
b & 0 & b & b & 0 \\
c & 0 & c & 0 & c
\end{tabular}
\end{center}
\vskip.1in

\noindent{\bf 4.} Find matrices $A$ and $C$ in $M(\R)$ such that $AC=\bf 0$ but $CA\neq \bf 0$, where $\bf 0$ is the zero matrix. [Hint: Example 6.]

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