The Relation Between Bézout Domain, Elementary Divisor Domain, and Adequate Domain

The aim of this paper is to investigate the relationship between Bézout domain, elementary divisor domain, and adequate domain. A Bézout domain is an integral domain D which every finitely generated ideal of D is principal. An integral domain D is called an elementary divisor domain if every matrix over D is equivalent to Smith normal form matrix. An adequate domain D is a Bézout domain and RP(a,b) exists for all a,b∈D with a≠0. Here the notion RP(a,b) defined as the relatively prime part of a with respect to b. It is found that every elementary divisor domain is a Bézout domain, but the converse is not true in general. It is shown the sufficient conditions for the Bézout domain being an elementary divisor domain. We also find out that every adequate domain is an elementary divisor domain. Furthermore, every one-dimensional Bézout domain is an adequate domain.