Filtration, asymptotic $\sigma$-prime divisors and superficial elements

Document Type : Research Paper


UFR Sciences Sociales, Universite Peleforo GON COULIBALY, Korhogo, Cote d'Ivoire


Let $(A,\mathfrak{M})$ be a Noetherian local ring with infinite residue field $A/ \mathfrak{M}$ and $I$ be a $\mathfrak{M}$-primary ideal of $A$. Let $f = (I_{n})_{n\in \mathbb{N}}$ be a good filtration on $A$ such that $I_{1}$ containing $I$. Let $\sigma$ be a semi-prime operation in the set of ideals of $A$. Let $l\geq 1$ be an integer and $(f^{(l)})_{\sigma} = \sigma(I_{n+l}):\sigma(I_{n})$ for all large integers $n$ and
$\rho^{f}_{\sigma}(A)= min \big\{ n\in \mathbb{N} \ | \ \sigma(I_{l})=(f^{(l)})_{\sigma}, for \ all \ l\geq n \big\}$. Here we show that, if $I$ contains an $\sigma(f)$-superficial element, then $\sigma(I_{l+1}):I_{1}=\sigma(I_{l})$ for all $l \geq \rho^{f}_{\sigma}(A)$. We suppose that $P$ is a prime ideal of $A$ and there exists a semi-prime operation $\widehat{\sigma}_{P}$ in the set of ideals of $A_{P}$ such that $\widehat{\sigma}_{P}(JA_{P})=\sigma(J)A_{P}$, for all ideal $J$ of $A$. Hence $Ass_{A}\big( A / \sigma(I_{l}) \big) \subseteq Ass_{A}\big( A / \sigma(I_{l+1}) \big)$, for all $l \geq \rho^{f}_{\sigma}(A)$.