Let R be a PID that is not a eld and let p R Show that p is

Let R be a PID that is not a eld, and let p R. Show that (p) is maximal if and only if p is irreducible.

Solution

Let R, p be as given.

Let (p) be maximal . If p is not irreducible , then there exist x ,y both not units such that

                         p=x y.

Let a be in (p). Then a =bp =bxy.

This implies a is in (x).

Thus (p) is contained in (x) , which contradicts the hypothesis that (p) is maximal.

Hence p is irreducible.

On the other hand, if p is not irreducible, then there exist x and y both not units such that

p=xy.

This would imply that (p) is contained in (x) , so p is not maximal.

Hence the result