Integral domains satisfy strict inclusions Euclidean ⊂ PID ⊂ UFD ⊂ (ID), with e.g. ℤ[(1+√−19)/2] a PID but not Euclidean, and ℤ[x] a UFD but not a PID.
Integral domains satisfy strict inclusions Euclidean ⊂ PID ⊂ UFD ⊂ (ID), with e.g. ℤ[(1+√−19)/2] a PID but not Euclidean, and ℤ[x] a UFD but not a PID.