ON SOME NON-EUCLIDEAN PRINCIPAL IDEAL DOMAINS
C. OBI MARTINS
Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria
C. ASHARA PRECIOUS
Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria
*Author to whom correspondence should be addressed.
Abstract
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).
Keywords: Euclidean domains, principal ideal domains, complex quadratic fields, algebraic integers, ring of integral algebraic integers in complex quadratic fields, field norm