Abstract

It is shown that if b 1,…, b s is a u-essential sequence over an ideal I in at Noetherian ring R, then certain permutations of u, tb 1,…, tb i , b i+1 ,…, b s are a u-esential sequence of IR(R,B i) and over tIR(R, I+B i), where B i = ( b 1, h., b i ) R (0 ⩽ i ⩽ s) andR(R, J) is the Rees ring of R with respect to its ideal J. A number of related results are also given concerning: u-essential sequences over I and over ID or I/b k ) D, where D is the monadic transformation ring R[ B i / b k ] or R[( I + B i )/ b k ]; when R is local, the u-essential cograde of I; a containment relation between the essential prime divisors of B i and the u-essential prime divisors of I + B i ; and the fact that every permutation of b 1,…, b s is a u-essential sequence over I and is an essential sequence in R.

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