Essential prime divisors and projectively equivalent ideals (Q579326)

In a series of papers the authors have established two new theories of ideals in Noetherian rings. The ``classical'' notions of associated prime divisor, R-sequence, Cohen-Macaulay ring are replaced by asymptotic (essential) prime divisors, asymptotic (essential) sequence, locally quasi-unmixed (unmixed) rings. We quote from the authors' introduction: ``Let I be an ideals in a Noetherian ring R. We concern ourselves with the essential prime divisors of I, an interesting subset of \(Ass(R/I^ n)\), for all large n. We first take \(I=bR\) with b a regular element of R. We show that there is a ring T, with \(R\subseteq T\subseteq R_ b\), such that T is a finite R-module and the essential primes of bT are exactly the prime divisors of bT. We next consider an arbitrary ideal I, and apply our principal arguments to the element u in the Rees ring of I. We thereby deduce that there is an ideal J projectively equivalent to I, such that the set of essential primes of I equals the set \(\cup Ass(R/J^ n),\) over \(n=1,2,3,....''\) One should note that the authors suggest a change in terminology in the appendix of this article.