Abstract:
In this paper, we introduce and study a class of integral domains D characterized by the property that whenever r; s 2 D ¡ f0g and the ideal (r k ; sk) is principal for some k 2 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 2 D for all u 2 D and k 2 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 2 D ¡ f0g and k 2 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 ?eld is a Good domain.
Page(s):
69-73
DOI:
DOI not available
Published:
Journal: Punjab University Journal of Mathematics, Volume: 46, Issue: 2, Year: 2014