ON A SUBCLASS OF SEMISTAR GOING-DOWN DOMAINS

Year 2013, Volume: 14 Issue: 14, 53 - 68, 01.12.2013

Parviz Sahandi Nematollah Shirmohammadi

Abstract

Let D be an integral domain and let ? be a semistar operation on
D. In this paper, we define the class of ?-quasi-going-up domains, a notion
dual to the class of ?-going-down domains. It is shown that the class of ?-quasi-going-up
domains is a proper subclass of ?-going-down domains and that every
Prüfer-?-multiplication domain is a ?-quasi-going-up domain. Next, we prove
that the ?-Nagata ring Na(D, ?), is a quasi-going-up domain if and only if D is
a e?-quasi-going-up and a e?-quasi-Prüfer domain. Several new characterizations
are given for ?-going-down domains. We also define the universally ?-going-down
domains, and then, give new characterizations of Prüfer-?-multiplication
domains.

Keywords

Semistar operation, integral domain, B´ezout domain, going-down domain, going-up, lying-over, quasi-going-up, quasi-going-up domain