Thread: Prime elements and maximal ideals in a PID

Here is the problem I have:

Let be a PID and . Prove that the ideal is maximal in if and only if is a prime element in . (Recall is a prime in if is not a unit and if then or )

I know that prime elements generate prime ideals (and I think I can prove that if I have to) and the only trick I had up my sleeve was that prime ideal -> ID and finite ID's are fields -> maximal, but it's not finite, so I am having difficulties constructing a proof...

Hi

One way is easy:



Now assume prime. Take i.e. classes of elements which are not in Since is an integral domain,
So imagine there is a such that
Is there a divisibility relation between and ? Then think of what that gives you in Using the fact that is a PID, conclude!

This is very much like what I managed to come up with. I finally broke down and showed that prime elements iff prime ideals, and then went that route.

The only tricky part was getting the two different divisibility cases for prime->maximal. One was obvious containment, thus maximal, and the other showed that the multiple was a unit, and thus containment.

Thanks a bunch!