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It is usual to prove that every Euclidean domain (ED) is a principal ideal domain (PID). This work developed and used inequalities to show that every Euclidean domain (ED) is a principal ideal domain and that the converse does not hold. It shows how the field norm may be applied to prove a simple result about the ring R of algebraic integers in complex quadratic fields Q⌊ √-M ⌋ which are Euclidean domains (EDs) and principal ideal domains (PIDs). Finally, how universal side divisors may be applied to prove some results about principal ideal domains (PIDs) which are not Euclidean domains (non-EDs).