S-Noetherian spectrum condition

A commutative ring R with identity satisfies the S-Noetherian spectrum condition, where SR is a given multiplicative set, if for each ideal I of R, for some sS and some finitely generated ideal J. Using this concept, we give an S-version of several different known results. For instance, the ring R satisfies the S-Noetherian spectrum property if and only if the polynomial ring R[X] satisfies the S-Noetherian spectrum property.