Abstract
Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a nonzero finitely generated [Formula: see text]-module and [Formula: see text] an ideal of [Formula: see text]. First purpose of this paper is to show that the sequences [Formula: see text] and [Formula: see text], [Formula: see text] 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 [Formula: see text] is included. A second purpose of this paper is to prove that [Formula: see text] has linear growth primary decompositions for Ratliff–Rush closures with respect to [Formula: see text], that is, there exists a positive integer [Formula: see text] such that for every positive integer [Formula: see text], there exists a minimal primary decomposition [Formula: see text] in [Formula: see text] with [Formula: see text], for all [Formula: see text].
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