Abstract. Let D be an integral domain and S be a multiplicative subset of D. Then given a semistar operation on D, we introduced the S-˜ -Noetherian domains, where˜ is the stable semistar operation of finite type associated to . Among other things, we provide many different characterization for S-˜ -Noetherian domains by focusing on primary decomposition, weak Bourbaki associated primes and Zariski-Samuel associated primes of the S-saturation of a given quasi-˜ -ideal I of D.Mathematics Subject Classification (2010): 13A15, 13F05, 13G05

Abstract. The purpose of this paper is to introduce a new class of rings that is closely related to the classes of pseudo valuation rings (PVRs) and pseudo-almost valuation domains (PAVDs). A commutative ring R is said to be a φ-ring if its nilradical Nil(R) is both prime and comparable with each principal ideal. The name is derived from the natural map φ from the total quotient ring T(R) to R localized at Nil(R). A prime ideal P of a φ-ring R is said to be a φ-pseudo-strongly prime ideal if, whenever x, y ∈ R Nil(R) and (xy)φ(P ) ⊆ φ(P ), then there exists an integer m 1 such that either x m ∈ φ(R) or y m φ(P ) ⊆ φ(P ). If each prime ideal of R is a φ-pseudo strongly prime ideal, then we say that R is a φ-pseudo-almost valuation ring (φ-PAVR). Among the properties of φ-PAVRs, we show that a quasilocal φ-ring R with regular maximal ideal M is a φ-PAVR if and only if V = (M : M ) is a φ-almost chained ring with maximal ideal √ M V . We also investigate the overrings of a φ-PAVR.