σ-sporadic prime ideals and superficial elements

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}}.

σ-sporadic prime ideals and superficial elements Keywords:

Noetherian ring , Filtration , semi-prime operation , associated prime ideals , superficial elements