See part (1) here! Again, we will assume that is a PID and is a varibale over In this post, we will take a look at the maximal ideals of Let be a maximal ideal of By Problem 2, if then for some prime and some which is irreducible modulo If then for some irreducible element Before investigating maximal ideals of in more details, let’s give an example of a PID which is not a field but has a maximal ideal which is principal. We will see in Problem 3 that this situation may happen only when the number of prime elements of is finite.
Example 1. Let be a filed and put the formal power series in the variable over Let be a variable over Then is a maximal ideal of
Proof. See that and that is the field of fractions of Thus is a field and so is a maximal ideal of
Problem 3. Prove that if has infinitely many prime elements, then an ideal of is maximal if and only if for some prime and some which is irreducible modulo
Solution. We have already proved one direction of the problem in Problem 1. For the other direction, let be a maximal ideal of By the first case in the solution of Problem 2 and the second part of Problem 1, we only need to show that So suppose to the contrary that Then, by the second case in the solution of Problem 2, for some We also know that is a field because is a maximal ideal of Since has infinitely many prime elements, we can choose a prime such that does not divide the leading coefficient of Now, consider the natural ring homomorphism Since and so is invertible in Therefore for some Hence for some If then we will have which is non-sense. So for some where does not divide the leading coefficient of Now gives us and so the leading coefficient of is divisible by Hence the leading coefficient of must be divisible by contradiction!
Example 2. The ring of integers is a PID and it has infinitely many prime elements. So, by Problem 3, an ideal of is maximal if and only if for some prime and some which is irreducible modulo By Problem 2, the prime ideals of are the union of the following sets:
1) all maximal ideals
2) all ideals of the form where is a prime
3) all ideals of the form where is irreducible in