POWERFUL IDEALS, STRONGLY PRIMARY IDEALS, ALMOST PSEUDO-VALUATION DOMAINS, AND CONDUCIVE DOMAINS

ABSTRACT

Let R be a domain with quotient field K, and let I be an ideal of R. We say that I is powerful (strongly primary) if whenever and , we have or ( or for some ). We show that an ideal with either of these properties is comparable to every prime ideal of R, that an ideal is strongly primary it is a primary ideal in some valuation overring of R, and that R admits a powerful ideal R admits a strongly primary ideal R is conducive in the sense of Dobbs-Fedder. Finally, we study domains each of whose prime ideals is strongly primary.

ACKNOWLEDGMENTS

The second author acknowledges support from NATO Grant 970140.