# Is every subring of a principal ideal ring also a principal ideal ring? [duplicate]

For the normal definition of a PIR (every ideal is principal in the ring), is every subring also a PIR?

I can't seem to think of a counterexample.

• $\mathbb Z[X]\subset\mathbb Q[X]$ – Pierre-Yves Gaillard Apr 10 '18 at 15:47
• @ArnaudD. is a PID equivalent to a PIR (just a difference of notation?) It may be the way I'm learning it in my class that is different – tdashrom Apr 10 '18 at 15:48
• @Pierre-YvesGaillard so does this not hold? proofwiki.org/wiki/Ring_of_Integers_is_Principal_Ideal_Domain – tdashrom Apr 10 '18 at 15:51
• @tdashrom - $\mathbb Z$ is a PID but $\mathbb Z[X]$ is not. – Pierre-Yves Gaillard Apr 10 '18 at 15:53
• The Domain part in PID means that there are no zero divisors. PID is PIR + Domain. – user550675 Apr 10 '18 at 16:01

Certainly $\Bbb C[X]$ is a PID. But $\Bbb C$ contains subrings isomorphic to $\Bbb Q[Y_1,Y_2]$ which have non-principal ideals.