| 广义幂级数环的PP性质 |
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| 引用本文： | 刘仲奎. 广义幂级数环的PP性质[J]. 数学学报, 2001, 44(6): 977-982. DOI: cnki:ISSN:0583-1431.0.2001-06-001 |
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| 作者姓名： | 刘仲奎 |
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| 作者单位： | 西北师范大学数学系 |
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| 基金项目： | 国家自然科学基金资助项目(19501007);教育部高等学校骨干教师资助项目 |
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| 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，（Ｊ，≤）是严格全序半群．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．
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| 关 键 词： | 广义幂级数环 PP-环 弱PP-环 |
| 文章编号： | 0583-1431(2001)06-0977-06 |
| 修稿时间： | 1998-04-30 |
PP-Rings of Generalized Power Series |
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| LIU Zhong Kui. PP-Rings of Generalized Power Series[J]. Acta Mathematica Sinica, 2001, 44(6): 977-982. DOI: cnki:ISSN:0583-1431.0.2001-06-001 |
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| Authors: | LIU Zhong Kui |
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| Affiliation: | LIU Zhong Kui (Department of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. China) (E-mail: liuzk@nwnu. edu. cn) AHSAN Javed (Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia |
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| Abstract: | 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. |
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| Keywords: | Ring of generalized power series PP-ring Weakly PP-ring |
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