# Math Help - Commutative UFD but not a PID

1. ## Commutative UFD but not a PID

Hi guys,

I'm trying to understand an example of a commutative UFD that is not a PID.

Let k be a field and consider the ring of polynomials over k, k[x,y]. Assuming that this is a UFD, I'm trying to show that this is not a PID.

It was shown to me a while ago before I had any 'further' knowledge and looking back at my notes now, I have

"xR+yR is not a principal ideal and not free..."

2. $Q[x,y]$ is a commutative UFD that is not a PID. A polynomial ring over a field is always a UFD. But in Q[x,y], the ideal (x,y) is not principal. Thus, Q[x,y] is not a principal domain...