广义幂级数环的PP性质  

PP-Rings of Generalized Power Series

作　　者：刘仲奎[1] AHSAN Javed 

机构地区：[1]西北师范大学数学系,甘肃兰州730070 [2]国王法赫德石油与矿藏大学数学科学系沙特阿拉伯

出　　处：《数学学报（中文版）》2001年第6期977-982,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金，教育部高校骨干教师资助计划

摘　　要：作为幂级数环的推广,Ribenboim引入了广义幂级数环的概念.设R是有单位元的交换环,(S,≤)是严格全序幺半群.本文中我们证明了如下结果:(1)广义幂级数环[[RS,≤]]是PP-环当且仅当R是PP-环且B(R)的任意S-可标子集C在B(R)中有最小上界;(2)如果对任意s∈S都有0≤s,则[[RS,≤]]是弱PP-环当且仅当R是弱PP-环.我们还给出了一个例子说明交换的弱PP-环可以不是PP-环.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 (5, ≤) a strictly totally ordered monoid. We prove that (1) the ring [[RS≤_]] 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∈5, then the ring [[RS≤_]] is weakly PP if and only if R is weakly PP.

关 键 词：幂级数环 PP-环 弱PP-环 广义幂级数环 交换环 幺半群 

分 类 号：O153.3[理学—数学]