Question

Prove or disprove: Every PID is UFD.​

Answer 1

Z[x] is not a PID (e.g. the set of polynomials in Z[x] whose constant term is even is a non-principal ideal) but Z[x] is a UFD. To see this note that irreducible elements in Z[x] are either integers of the form ±p for a prime p, or primitive irreducible polynomials of degree ≥ 1.