I've run into this question that should be easy somehow, but I can't get a hold of:

Is R a PID?

Naturally, is a PID since it's a field. But since R isn't an ideal in , I can't use that in an argument in any way. (It's not, because 1/2*r with r in R isn't in it by default, and 1/2 is in ).

Not every element has a multiplicative inverse in R, since a can be negative, so it's also surely not a field (which would also imply it is a PID)...

So how do I go about it? R is isomorphic with . Would that form an argument somehow?

Also, can anyone confirm for me that R/2R is isomorphic to and R/3R is isomorphic to for me?

My argument is that you can make a hormomorphism f: by if a is even, and if a is odd. This way only all elements in 2R can be found in . So by the isomorphism theorem .

And the same argument applies for R/3R, by sending any r with a that is p mod 3 to p.