مشخصات پژوهش

Let R be a commutative Noetherian ring, E a nonzero finitely generated R-module and I an ideal of R. First purpose of this paper is to show that the sequences AssR E/fInE and AssR fInE /
In+1 E , n = 1, 2, . . . , of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff [On the relevant transform and the relevant component of an ideal, J. Algebra 111 (1987) 507–519, Theorem 3.1]. In addition, a characterization concerning the set fA∗(I,E) is included. A second purpose of this paper is to prove that I has linear growth primary decompositions for Ratliff–Rush closures with respect to E, that is, there exists a positive integer k such that for every positive integer n, there exists a minimal primary decomposition fInE = Q1 ∩ ·· ·∩Qs in E with (Rad(Qi :R E))nk ⊆ (Qi :R E), for all i = 1, . . . , s.