Field implies Euclidean implies PID implies UFD implies domain, with every implication strict.Suggest a slogan
Every field is a Euclidean domain, every Euclidean domain is a principal ideal domain, every principal ideal domain is a unique factorization domain, every unique factorization domain is an integral domain, and every integral domain is a commutative ring: \[ \text{Fields} \subset \text{Euclidean domains} \subset \text{PIDs} \subset \text{UFDs} \subset \text{Integral domains} \subset \text{Commutative rings}. \]
Each inclusion is proper: \(\ZZ\) is a Euclidean domain that is not a field; \(\ZZ\big[\tfrac{1+\sqrt{-19}}{2}\big]\) is a PID that is not Euclidean; \(\ZZ[x]\) is a UFD that is not a PID; \(\ZZ[\sqrt{-5}]\) is an integral domain that is not a UFD; and \(\ZZ/4\ZZ\) is a commutative ring that is not an integral domain.