Dummit and Foote, Section 8.2 (Principal Ideal Domains (PIDs) ) - Exercise 4, page 282.
Let R be an integral domain.
Prove that if the following two conditions hold then R is a Principal Ideal Domain:
(i) any two non-zero elements a and b in R have a greatest common divisor which can be written in the form ra + sb for some and
(ii) if are non-zero elements of R such that for all i, then there is a positive integer N such that is a unit times for all