Remark
The theorem generalises two known cases:
Ideals in Z
Ideals in polynomial rings
The ideal (n)
of Z is prime if and only if n is prime;
then Z/(n) is a field, so then (n) is even maximal.
Similar to the previous case,
each ideal of K[X], where K is a field,
is of the form (f) for some polynomial f.
The ideal is prime if and only if it is maximal, in which case f
is irreducible.