ON SOME NON-EUCLIDEAN PRINCIPAL IDEAL DOMAINS

Main Article Content

C. OBI MARTINS
C. ASHARA PRECIOUS

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

Article Details

How to Cite
MARTINS, C. O., & PRECIOUS, C. A. (2019). ON SOME NON-EUCLIDEAN PRINCIPAL IDEAL DOMAINS. Asian Journal of Mathematics and Computer Research, 25(8), 478-510. Retrieved from http://ikpress.org/index.php/AJOMCOR/article/view/4250
Section
Original Research Article