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{\Huge 189-235A: Basic Algebra I} \\ \vskip 0.1in
{\Huge Assignment 4} \\ \vskip 0.1in
{\Large Due: Friday, October 14 }
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1. Let $R$ be a commutative ring.
Is the set $S$ of matrices of the form
$\left(\begin{array}{cc}
a & b \\ 0 & c \end{array}\right)$,
with $a,b,c\in R$, a subring of $M_2(R)$?
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2. Same question as 1, but with $S$ replaced by the set of matrices
of the form
$\left(\begin{array}{cc}
a & b \\ 0 & a \end{array}\right)$,
with $a,b\in R$.
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3. Show that the ring $\Q$ of rational numbers
has no subrings which are finite sets.
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4. Let $A$ be a commutative ring, let $R=A\times A$,
and let $S$ be the subset of elements of $R$ of the form
$(a,0)$, with $a\in A$.
Show that $S$ is a ring which is isomorphic to $A$.
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5. Let $R$ be a ring, and let $a$
be an element of $R$ which is not a zero divisor.
Show that the cancellation law can be applied to $a$, i.e.,
for all $x,y\in R$,
if $ax=ay$ then $x=y$.
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6. Let $R=\Q(\sqrt{2})$ be the ring of elements of the form
$a+b\sqrt{2}$, with $a,b\in \Q$.
Show that the function which sends $a+b\sqrt{2}$ to
$a-b\sqrt{2}$ is an isomorphism from $R$ to itself.
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7. Let $R$ be a ring.
Show that there is a {\em unique} ring homomorphism $f$ from $\Z$ to
$R$.
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8. Given a ring $R$, let $R[i]$ denote the set of pairs $(a,b)$ with
$(a,b)\in R$. Define an addition and multiplication on
$R[i]$ by the rules:
$$ (a_1,b_1)+ (a_2,b_2) = (a_1+a_2, b_1+b_2).$$
$$ (a_1,b_1)(a_2,b_2) = (a_1 a_2 -b_1b_2, a_1b_2+b_1a_2).$$
Show that these rules equip $R[i]$ with the structure of a ring.
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9. Show that the subset $S$ of the ring $R[i]$ of exercise 8 consisting of
elements of the form $(r,0)$ with $r\in R$ is a subring of $R[i]$
which is isomorphic to $R$.
Produce an element $i$ in $R[i]$ satisfying $i^2=-1$.
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10. Show that the rings $\Z_{48}$ and $\Z_6\times\Z_8$ are {\em not}
isomorphic.
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{\bf Optional questions}.
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11. Keeping the notations of exercise 8,
show that the ring $\C[i]$ is isomorphic to $\C\times \C$.
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12. Show that a ring $R$ which is a finite set and an integral domain
(has no zero-divisors)
is necessarily a field.
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