The Good Property of Two-Generated Ideals in Integral Domains

Abstract

In this paper, we introduce and study a class of integral domains D characterized by the property that whenever r, s ∈ D − {0} and the ideal (rk, sk) is principal for some k ∈ N, then the ideal (r, s) is principal. We call them Good domains. We show that a Good domain D is a root closed domain and the converse is true in different cases as follows: (1) D is quasi-local, (2) P ic(D) = 0, (3) u1/k ∈ D for all u ∈ D and k ∈ N, (4) D is t-local. We also show that a quasi-local domain D with the property that (r, s) k = (rk, sk) for all r, s ∈ D − {0} and k ∈ N, is a Good domain, that a Prufer Good domain with torsion Picard ¨ group is a Be´zout domain, and that the integral closure of a domain in an algebraically closed field is a Good domain.