PP-Rings of Generalized Power Series

• DOI: 10.1007/s1011400000884
• Publication Date: 2007-02-28T18:58:29Z
• Authors: Zhongkui Liu; Javed Ahsan
• Keywords: 0101 mathematics; 01 natural sciences

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.