σ-sporadic prime ideals and superficial elements

Abstract

Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I)$, with $n$ small, which are the sets of $sigma$-sporadic prime divisors of $I$.We also give some relationships between $sigma(f_I)$-superficial elements and asymptotic prime $sigma$-divisors, where $sigma (f_I)$ is the $sigma$-closure of the $I$-adic filtration $f_I=(I^n)_{ninmathbb{N}}$.