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.