1. Isomorphisms
We return here to the quotient ring construction. Recall that if is a commutative ring with identity and
is an ideal of
then
is also a commutative ring with identity. Here,
where
for
the equivalence relation defined by
iff
Since is an equivalence relation, we have that
if
and
if
In particular, any two classes are either the same or else they are disjoint.
In case for some field
then
is principal, so
for some
i.e., given any polynomial
iff
and, more generally,
(or, equivalently,
or, equivalently,
) iff
In this case, contains zero divisors if
is nonconstant but not irreducible.
If is 0,
If is constant but nonzero, then
Finally, we want to examine what happens when is irreducible. From now on suppose that this is the case.