It is well-known that the rings O (d) of algebraic integers in 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 ae {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 GE (2)-rings.
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