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\centerline{Mathematics 351 -- Abstract Algebra I}
\centerline{Discussion 1 -- More Examples of Rings}
\centerline{September 3, 2007}
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\noindent
{\it Background}
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Last time, we introduced the general definition of a {\it ring},
an nonempty set $R$ with two binary operations $+,\cdot$ satisfying
the following 8 axioms:
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\item{1.}  (closure under $+$) For all $a,b \in R$, $a + b\in R$.
\item{2.}  (associativity of $+$)  For all $a,b,c \in R$, $(a + b) + c = a + (b + c)$.
\item{3.}  (commutativity of $+$)  For all $a,b \in R$, $a + b = b + a$.
\item{4.}  (additive identity)  There is an element $0 \in R$ such that $a + 0 = a$ for 
all $a \in R$. 
\item{5.}  (additive inverses)  For each $a \in R$, there is an element $x = -a \in R$ such
that $a + x = 0$.
\item{6.}  (closure under $\cdot$)  For all $a,b \in R$, $a\cdot b \in R$. 
\item{7.}  (associativity of $\cdot$) For all $a,b,c \in R$, $(a\cdot b)\cdot c = a\cdot (b\cdot c)$.
\item{8.}  (distributive laws)  For all $a,b,c \in R$, $a\cdot (b + c) = a\cdot b + a\cdot c$ {\it and}
$(a + b)\cdot c = a\cdot c + b\cdot c$.
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\noindent
If, in addition there is a multiplicative identity $1$ in $R$, we call $R$ a ring
{\it with identity}.  If the multiplication operation $\cdot $ is also commutative,
we say $R$ is a {\it commutative} ring.  

In a ring $R$, a {\it zero divisor} is an
element $a \ne 0 \in R$ such that there exists some $b \ne 0$ with $a\cdot b = 0$ or 
$b\cdot a = 0$.
An {\it integral domain} is a commutative ring with identity that has no 
zero divisors (i.e. where $a\ne 0$ and $b \ne 0$ implies $a\cdot b \ne 0$).

Finally, a commutative ring with identity $1 \ne 0$, in which every nonzero element has
a multiplicative inverse (that is, for each $a \ne 0$, the 
equation $a\cdot x = 1$ has a solution $x = a^{-1}$ in the ring) is called a {\it field}.

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\noindent
{\it Discussion Questions}
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\noindent
I.  Let ${\bf Q}$ denote the field of rational numbers, and let $p$ be 
a (fixed) prime number in ${\bf Z} \subset {\bf Q}$.  Let 
$$R_p = \left\{ {a \over b} \in {\bf Q} : a,b \in {\bf Z} {\rm\ and\ } b = p^i {\rm\ for\ some\ } i\ge 0\right\}.$$
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\item{A)} Show carefully that $R_p$ is a commutative ring with identity, using 
the usual sum and product operations from ${\bf Q}$.  {\bf Note}:  Many of the 
ring properties in $R_p$ will follow from corresponding statements in ${\bf Q}$,
because $R_p$ is a subset of the ring ${\bf Q}$,
so if you are thinking about it the right way, 
this should not be as tedious as it might appear at first(!)
\item{B)}  Is $R_p$ an integral domain?  Is $R_p$ a field?  Prove your assertions.
\item{C)}  The ring $R_p$ here is an example of what is called a {\it subring} of the 
ring ${\bf Q}$.  What should it mean to say a ring $S$ is a subring of a ring $R$ in general?
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\noindent
II.  Let $L = \{a \in {\bf R} : a > 0\}$.  We can define new
``sum'' and ``product'' operations on $L$ as follows:
$$a \oplus b = ab \qquad {\rm and} \qquad a \otimes b = a^{\ln(b)}.$$
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Is $L$ is a ring under these operations?  Why or why not?  Is it a {\it subring}
of the ring $({\bf R},+,\cdot)$ ($+,\cdot = $ the usual sum and product of reals)? 
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\noindent
III.  In class last time, we saw that 
$$C^{1}({\bf R}) = \{f : {\bf R} \to {\bf R}:  f' {\rm\ exists\ and\ is\ continuous\ at\ all\ } x \in R\}$$
is a commutative ring with identity.
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\item{A)}  Let
$$f(x) = \cases{3x^2 - 2x^3 & if $0 \le x \le 1$\cr
                2x^3 - 9x^2 + 12x - 4 & if $1 \le x \le 2$\cr
                0 & otherwise\cr}$$
Sketch the graph of $f$ and show that $f$ is an element of $C^{1}({\bf R})$.  
\item{B)}  Is $C^1({\bf R})$ an integral domain?  ({\bf Hint:}  You can make lots
of other functions in $C^1({\bf R})$ by ``doing things to'' the function $f$
from part A.)
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\noindent
IV.  Let $(R,+_R,\cdot_R),(S,+_S,\cdot_S)$ 
be any two rings.  We can define sum and product operations $+, \cdot$
``component-wise'' on the Cartesian product:
$$R \times S = \{(a,b) : a \in R, b \in S\}$$
by the following rules:
$$\eqalign{
(a,b) + (a',b') &= (a +_R a', b +_S b')\cr
(a,b) \cdot (a',b') &= (a \cdot_R a', b \cdot_S b').\cr}$$
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\item{A)}  Show that the above operations always make $R\times S$ into
a ring.
\item{B)}  Let $R = {\bf Z}_2$ with addition and multiplication mod 2,
and let $S = {\bf Z}_3$ with addition and multiplication mod 3.
Make addition and multiplication tables for the operations $+, \times$
on ${\bf Z}_2\times {\bf Z}_3$ defined as above.
\item{C)}  Is ${\bf Z}_2\times {\bf Z}_3$ ``the same'' as some other ring
we know?  In what sense?
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\noindent
{\it Assignment}
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Group writeups due on Monday, September 10.
\bye
