σ-SPORADIC PRIME IDEALS AND SUPERFICIAL ELEMENTS

Let a be a noetherian ring, i be an ideal of a and σ 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 σ(i n), which denoted by ass(a/σ(i n), stabilize to aσ(i). we give some properties of the sets s σ n (i) = ass(a/σ(i n)aσ(i), with n small, which are the sets of σ-sporadic prime divisors of i. we also give some relationships between σ(fi )- superficial elements and asymptotic prime σ-divisors, where σ(fi ) is the σ-closure of the i-adic filtration fi = (i n)n∈n.