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 (r
k
, 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) u
1/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 = (r
k
, 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.
Waseem Khalid, Fakhira Umar. (2014) The Good Property of Two-Generated Ideals in Integral Domains, Punjab University Journal of Mathematics, Volume 46, Issue 2.
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