PP-Rings of Generalized Power Series

As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S, .) a strictly totally ordered monoid. We prove that (1) the ring [[R S,.]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R) and (2) if (S, .) also satisfies the condition that 0 .s for any s.S, then the ring [[R S,. ]] is weakly PP if and only if R is weakly PP.