Let be a nontrivial prime ideal, and let be an Ideal containing
So we have Now for P to be maximal either or
Since I is generated by some element a of the Ring
Now so for some Now since P is a prime Ideal and either or .
So we have two cases.
Case I then .
Case II then
now we know that as well, so we sub this into above to get
and Since we are in an integral domain and we know that
so are both units
so this implies that and we are done