ON SOME NON-EUCLIDEAN PRINCIPAL IDEAL DOMAINS |Asian Journal of Mathematics and Computer Research

Every Euclidean domain (ED) is usually proved to be a principal ideal domain (PID). This work constructed and applied inequalities to demonstrate that every Euclidean domain (ED) is a principal ideal domain, despite the fact that the opposite is not true. It demonstrates how to use the field norm to prove a basic fact 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 can be used to prove various statements regarding non-Euclidean principal ideal domains (PIDs) (non-EDs).

Please see the link :- https://www.ikprress.org/index.php/AJOMCOR/article/view/4250