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{\bf Math 222: A Brief Introduction to Rings}

We have discussed two fundamental algebraic structures: {\em fields} and
{\em groups}.  A field is a world with two operations
(addition and multiplication) which satisfy all the properties we're used
to.  A group has only one operation which need not be
commutative.  Somewhere in between these two worlds is a third
fundamental structure: a {\em ring}.

{\bf Definition.} A {\em ring} is a set $R$ with two operations
$(+,\cdot)$ such that $R$ forms an abelian group under addition
and the multiplication is associative and distributes over addition.
More precisely, for all $a, b, c \in R$, the following must hold:
\begin{itemize}
\item[(1)] $a + b = b + a$.
\item[(2)] $(a + b) + c = a + (b + c)$.
\item[(3)] There is an element $0$ in $R$ such that $a + 0 = a$.
\item[(4)] There is an element $-a$ in $R$ such that $a + (-a) = 0$.
\item[(5)] $a(bc) = (ab)c$.
\item[(6)] $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$.
\end{itemize}

Clearly any field satisfies all these properties and so is
a ring.  But, let's look at what properties are NOT assumed for
a ring:
\begin{itemize}
\item The multiplication is not assumed to be commutative.  If it is,
the ring is said to be {\em commutative}.  Note: We do not say that a
ring is {\em abelian} -- that terminology is reserved for groups.
\item The ring need not have a $1$, that is a multiplicative identity
element.  If it does, we say the ring has a {\em unity} or has an {\em
identity} or has a {\em one} or the ring is {\em unital}. If a ring does
have a unity, the unity is  unique.
\item Even if the ring does have a unity, there is no assumption that
multiplicative inverses exist.  An element of a unital ring which does
admit a multiplicative inverse is called a {\em unit}.  If they exist,
inverses are also unique in a ring.  Further, the set of units in a ring
forms a group under multiplication -- the {\em unit group} of the ring.
\end{itemize}

In this language, a field is a commutative ring with unity in which
every non-zero element is a unit. Besides fields, we have already come
across many rings in this course:

{\bf Example 1.} The integers $\zz$ under usual addition and
multiplication is a commutative ring with unity -- the unity
being the number 1.  Of course the only units are $\pm 1$.

{\bf Example 2.} For any positive integer $n > 0$, the integers mod $n$,
$\zz_n$, is a commutative ring with unity.  We have seen that the units
are those elements which are relatively prime to $n$. The unit group is
denoted $U(n)$. Of course, if $n$ happens to be a prime, then we have a
field.

{\bf Example 2.1} Consider $\zz_{10}$.  The group of units is
$U(10) = \{1, 3, 7, 9\}$.  Which group of order 4 is this isomorphic to?
We have already noticed something ``bad'' which can happen in rings:
both 2 and 5 are non-zero elements of $\zz_{10}$ and yet $2\cdot5 = 0$
mod 10.  The numbers 2 and 5 are called {\em zero-divisors}.  In a field
zero is the only zero-divisor.  A commutative ring with unity which has
NO non-zero zero-divisors is called an {\em integral domain}.  For
example,
$\zz$ is an integral domain as is any field.

{\bf Example 3.} Given any ring $R$, the set of polynomials with
coefficients in $R$:
$$
R[x] = \{a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0: a_i \in R\}
$$
under polynomial addition and multiplication is a commutative ring with
unity (the constant polynomial 1).  We have already observed that most
polynomials do not have an inverse.  If $R$ has no (non-zero)
zero-divisors, then the polynomial ring over $R$, $R[x]$, has no
(non-zero) zero-divisors.

{\bf Example 3.1.} What are the units in $\mathbb{R}[x]$? $\zz[x]$?

{\bf Example 4.} The set of even integers $2\zz \subset \zz$ is a ring.
Indeed, it is a {\em subring} of $\zz$ and is of course commutative.
Does it have a unity?

{\bf Example 5.} An important family of non-commutative rings are
{\em matrix rings}.  For example, consider the set of all 2 $\times$ 2
matrices with real entries:
$$
M_2(\mathbb{R}) = \Big{\{}
\begin{pmatrix}
a & b\\
c & d\\
\end{pmatrix}: a, b, c, d \in \mathbb{R}\Big{\}}.
$$
This is a ring under matrix addition and multiplication.  It has an
identity -- what is it?  But, it is not commutative.  Can you give an
example?  What is the group of units?  Note: One can consider matrices
of arbitrary size and having entries in any field or ring.

%{\bf Some Terminology:}

{\bf Definition.} Two rings $R$ and $S$ are said to be {\em isomorphic}
if there is a map $f: R \to S$ satisfying the following three properties:
\begin{itemize}
\item $f$ is one-to-one: For $a, b \in R$, if $f(a) = f(b)$, then $a =
b$.
\item $f$ is onto: For each $b \in S$, there is $a \in R$ with $f(a) =
b$.
\item $f$ is a {\em homomorphism of rings}: For all $a, b \in R$,
$f(a + b) = f(a) + f(b)$ and $f(a\cdot b) = f(a)\cdot f(b)$.
\end{itemize}

{\bf Example 6.} Consider the fields $F = GF(3,x^2 + x + 2)$ with Galois
imaginary $\alpha$ and $F' = GF(3,x^2 + 2x + 1)$ with Galois imaginary
$\beta$.  Both $\alpha$ and $\beta$ are primitive elements in their
respective fields and the map $f: F \to F'$ by $f(\alpha^n) = \beta^n$
for each $0 \leq n \leq 8$ is an isomorphism of rings. Or in this case
we would say an isomorphism of fields.

{\bf Example 6.1} The map $f: \mathbb{Z} \to 3\mathbb{Z}$ by $f(x) = 3x$
is an isomorphism of groups (under the addition structure) but is NOT an
isomorphism of rings since $f(2\cdot 2) \neq f(2)\cdot f(2)$.

{\bf Definition.} A subring $S \subset R$ a ring is said to be a
(two-sided) {\em ideal} if for every $s \in S$ and $r \in R$, both $rs$
and $sr$ are in $S$.

{\bf Example 7.} The subring $2\zz \subset \zz$ is an ideal.  For an
element of $2\zz$ is even and multiplying by any other integer, it
remains even.  Moreover, for any postive integer $n$, the subset $n\zz
\subset \zz$ is an ideal.

{\bf Example 8.} Consider the polynomial ring $\mathbb{R}[x]$ and let
$S$ be the subset of all polynomials with zero constant term.  This is a
subring and in fact an ideal.  It is usually denoted $\langle x\rangle$
and called the ideal generated by $x$.  The point being that every
polynomial with  zero constant term admits $x$ as a factor; in other
words, is a multiple of $x$.

{\bf Example 8.1.} Consider the ideal $\langle x^2\rangle \subset
\mathbb{R}[x]$.  Can you figure out what this set is?  In particular,
which of the following polynomials lie in $\langle x^2\rangle$:
$$
x^5 + 2x^3 - x^2,~~~~ x^2 + 2,~~~~ 5x^3 + x,~~~~ 10x^{35} - x^{16} ~~?
$$

This notion of an ideal allows one to form so called {\em factor rings}
by ``dividing'' in some sense a ring by an ideal.  This is similar to
the notion of a quotient group.
\end{document}