Ratliff-Rush closures and linear growth of primary decompositions of ideals

Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\Ass_RE/ \widetilde{I_E^n}$ and $\Ass_R\widetilde{I^n _E}/\widetilde{I^{n+1}_E}$, $n = 1,2, \dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\widetilde{A^*}(I,E)$ 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 $\widetilde {I^n_E}= Q_1\cap \cdots\cap Q_s$ in $E$ with $(\Rad(Q_i:_RE))^{nk}\subseteq (Q_i:_R E)$, for all $i= 1, \dots, s$.