Domains whose ideals meet a universal restriction

Abstract

Let [Formula: see text] represent a set of proper nonzero ideals [Formula: see text] (respectively, [Formula: see text]-ideals [Formula: see text]) of an integral domain [Formula: see text] and let [Formula: see text] be a valid property of ideals of [Formula: see text] We say [Formula: see text] meets [Formula: see text] (denoted [Formula: see text] if each [Formula: see text] is contained in an ideal satisfying [Formula: see text]. If [Formula: see text] [Formula: see text] [Formula: see text] cannot be controlled. When [Formula: see text] [Formula: see text] [Formula: see text] does not imply [Formula: see text] [Formula: see text] while [Formula: see text] [Formula: see text] implies [Formula: see text] [Formula: see text] usually. We say [Formula: see text] meets [Formula: see text] with a twist [Formula: see text]written [Formula: see text] if each [Formula: see text] is such that, for some [Formula: see text] [Formula: see text] is contained in an ideal satisfying [Formula: see text] and study [Formula: see text] as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.