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\begin{center}
{{\Large\bf MODERN ALGEBRA I}\\
\bigskip

{\Large\bf HOMEWORK ASSIGNMENT VII}\\
\bigskip
{\bf (Turn in underlined problems.)}}
\end{center}

\vspace{.75in}

\noindent
{\bf 
\begin{tabbing}
\underline{\large\bf CHAPTER}\makebox[2in]{}\=\underline{\large\bf PROBLEMS}\\

\bigskip
\noindent
3 \> \underline{3.23},
\underline{3.24(a)}
\end{tabbing}

}
\vspace{.75in}

\noindent
\underline{\large\bf ADDITIONAL PROBLEMS}
\vspace{.25in}

{\bf 

\underline{1.}
Define an equivalence relation on points in the
Euclidean plane ${\BR}^2$ by 
saying that $(a,b)\equiv (c,d)$ if 
$(a,b)$ and $(c,d)$ have the same distance
from $(0,0)$.
Describe the equivalence classes of this equivalence
relation.

\bigskip

\underline{2.}
Let {\bf S} be a commutative ring.
Define a relation on $S$
by $a\equiv b$ if there is a unit $u$
such that $a = ub$. Show that $\equiv$
is an equivalence relation.

\bigskip

\underline{3.}
Show that the set of all rational
numbers $m/n$,
where $n$ is odd, is a subring of $\BQ$.

\bigskip

4. Let $P = \{p_1, p_2, \ldots\}$ be a set
of \underline{some} of the primes.
Show that the set of all rational numbers of the
form $m/n$,
where $n$ is not divisible by any prime from $P$,
is a subring of $\BQ$.

\bigskip

5. Show that every subring of $\BQ$ is of
the type described in Problem 2.

}
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