tλESSENTIAL PRIME DIVISORS AND SEQUENCES OVER AN IDEAL

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian. In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, i?-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasiunmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than i?-sequences (for example, they behave nicely when passing to Rjz with z a minimal prime ideal in i?), the converse of this working guide has also proved useful. However, in a number of problems it has turned out that the asymptotic theory is a little too coarse, so it seemed worthwhile to try to develop a new theory that behaved nicely when passing to R\z with z an arbitrary prime divisor of zero (rather than just a minimal prime divisor of zero). Such a theory would then be intermediate between the standard and asymptotic theories, and would thereby surmount some of the problems encountered in the asymptotic theory. One candidate for this new intermediate theory was developed in [7], where it was called the "essential"