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. Let (R,m) be a Noetherian local ring, I an ideal of R, and Aan Artinian R-module. Let k≥0 be an integer and r= Width >k (I,A)the supremum of length of A-cosequence in dimension > k in I denedby Nhan-Hoang [8]. It is shown that for all t≤r the sets([ ti=0 Att R (Tor Ri (R/I n ,A))) ≥k and([ ti=0 Att R (Tor Ri (R/(a n 1 1 ,...,a l l ),A))) ≥k are independent of the choice of n,n 1 ,...,n l for any system of generators(a 1 ,...,a l ) of I. 1. IntroductionThroughout this paper, let (R, m ) be a Noetherian local ring, I an ideal ofR. Let M be a nitely generated R-module, and Aan Atinian R-module. Fora subset T of Spec(R) and an integer i≥ 0, denote by (T) ≥i the set of all primeideals p ∈ T such that dimR/ p ≥ i.In 1976, L. J. Ratli [10] asked whether the set of associated prime idealsAss R (R/I n ) is stable for large n. An armative answer was given by Ratli[10] for the case R is an domain. M. Brodmann [1] answered fully Ratli’squestion not only for ring R, but also for module M. He proved that the setsAss

AN INDEPENDENT RESULT FOR ATTACHED PRIMES OF CERTAIN TOR-MODULES