Answer to Question #12541 in Algebra for Hym@n B@ss

Question

Prove that K[X] over integral domain K is principal ideal ring iff K is a field

Answer 1

Let we have K[X] - PID. Then for any a<>0 we consider ideal
I=<a,X>.
As it is PID we have b in K that I=<b>.
X belongs to
I then X=(dX)b. Then db=1. Hence b=d^(-1) => I=<b>=K[X].
So, 1=
u(X)X+v(X)a. => 1=v(0)a => a - invertible => K is
field.
Reverse implication is obvious.