PP-Rings of Generalized Power Series

摘要

As a generalization of power series rings, Ribenboim introduced the notion of the rings ofgeneralized 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-ringand every S-indexed subset C of B（R） （the set of all idempotents of R） has a least upper bound inB（R）. and （2） if （S. ≤） also satisfies the condition that 0≤s for any s∈S, then the ring [[R(S.≤]] isweakly PP if and only if R is weakly PP.