A remark on the ring of algebraic integers in Q(√-d)

It is well-known that the rings O_d of algebraic integers in (√-d) for d = 19, 43, 67, and 163 are principal ideal domains but not Euclidean. In this article we shall provide a method, based on a result of P. M. Cohn, to construct explicitly pairs (b, a) of integers in O_d for d = 19, 43, 67, and 163 such that, in O_d, there exists no terminating division chain of finite length starting from the pairs (b, a). That is, a greatest common divisor of the pairs (b, a) exists in O_d but it can not be obtained by applying a terminating division chain of finite length starting from (b, a). Furthermore, for squarefree positive integer d ∉ {1, 2, 3, 7, 11, 19, 43, 67, 163}, we shall also construct pairs (b, a) of integers in O_d which generate O_d but have no terminating division chain of finite length. It is of interest to note that our construction provides a short alternative proof of a theorem of Cohn which is related to the concept of GE₂-rings.