σ-sporadic prime ideals and superficial elements

Document Type: Research Paper

Authors

1 D'epartment de Sciences et Technologie, Section Math'ematiques, Ecole normale sup'erieure, Abidjan, C^ote d'Ivoire

2 UFR sciences sociales, Universit'e P'el'eforo Gon Coulibaly, Korhogo, C^ote d'Ivoire

3 Laboratoire de Math'ematiques et Informatique, Universit'e Nangui Abrogoua, Abidjan, C^ote d'Ivoire

4 UFR de Math'ematiques et Informatique, Universit'e F'elix Houphouet Boigny, Abidjan, C^ote d'Ivoire

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 $f_I=(I^n)_{n\in\mathbb{N}}$.

Keywords