PF-rings of Generalized Power Series

Abstract. In this paper, we show that if R is a commutative ring with identity and ( S;• )is a strictly totally ordered monoid, then the ring [[ R S;• ]] of generalized power series is aPF-ring if and only if for any two S -indexed subsets A and B of R such that B µ ann R ( A ),there exists c 2 ann R ( A ) such that bc = b for all b 2 B , and that for a Noetherian ring R ,[[ R S;• ]] is a PP ring if and only if R is a PP ring. 1. Introduction and preliminariesLet R be a commutative ring with identity. Then R is called a PF-ring (resp., PP-ring ) if every principal ideal of R is a flat (resp., projective) R -module. It iswell-known that if R is Noetherian, then these two notions are equal (cf., [16, Corol-lary 4.3]). It is proved in [1] that a ring R is a PF-ring if and only if the annihilatorof each element r 2 R , ann R ( r ), is a pure ideal; that is, for all b 2 ann R ( r ) thereexists c 2 ann R ( r ) such that bc = b . It may be worth reminding the reader that fora commutative ring