Non-commutative Valuation Rings and Applications to their Global Theories Quantum Groups

非交换估值环及其全局理论量子群的应用

• 批准号: 09640044
• 负责人: MARUBAYASHI Hidetoshi
• 金额: $ 0.96万
• 依托单位: Naruto University of Education
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640044/
• 关键词: Valuation rings; Prime ideals; Primary ideals; Skew group rings; Gabriel topologies; Prufer; Semihereditary; Overrings; cycles; total valuation; 整環; localization; skew polymomial; principal ideal rings; Jacobson radical; Skew growp rings

The main outcomes are classified into the following three :1.The classifications of prime and primary ideals in Dubrovin valuation rings.We can classified the set of prime ideals in Dubrovin valuation rings into four different types by using the cocepts of prime segments and of limit primes. We found that there were five different types of primary ideals. These classifications are applied for the classifications of prime and primary ideals in P.I.Prufer orders. In addition to this, we also applied the above results to classification of Gabriel topologies on Dubrovin valuation rings.2.We could find a necessary and sufficient conditions for the skew group rings to be semi-hereditary and Prufe by using the properties of the coefficient ring and goup.3.We studied the structure of semihereditary orders which are integral ove commutative valuation rings. We knew that there were five different type of cyles among the set of all maximal ideals. With these cycles's properties, we could investigate the exact numbers of overring of a given semi-hereditary order and the properties of the Jacobson radical. Furthermore, we found a lot of semi-hereditary maximal orders which are not prufer orders.

1. Dubrovin赋值环中素理想和准素理想的分类利用素段和极限素的概念，我们可以将Dubrovin赋值环中的素理想集分为四类。我们发现有五种不同类型的基本理想。这些分类被应用于P.I.Prufer序中素理想和准素理想的分类。此外，还将上述结果应用于Dubrovin赋值环上Gabriel拓扑的分类; 2.利用系数环和群的性质，得到了斜群环是半遗传的和Prufe的充要条件; 3.研究了交换赋值环上半遗传序的结构.我们知道在所有极大理想的集合中有五种不同类型的圈。利用这些圈的性质，我们可以研究给定半遗传序的上环的确切个数以及Jacobson根的性质。此外，我们还发现了许多非Prufer序的半遗传极大序。

项目成果

丸林英俊: "Skew group rings which are semi-hereditary orders and Prufer orders"Algebra and Representation Theory. 3. 259-275 (2000)

Hidetoshi Marubayashi：“半遗传阶和普鲁弗阶的斜群环”代数和表示理论 3. 259-275 (2000)。

丸林英俊: "A cleasification of prime segments in simple rings"Proc.A.M.S.. (accepted).

Hidetoshi Marubayashi：“简单环中素数段的净化”Proc.A.M.S.（已接受）。

H.Marubayashi: "Skew group rings which are semi-hereditary and Prufer orders"Algebra and Representation Theory. 3. 259-275 (2000)

H.Marubayashi：“半遗传性和普鲁弗阶的斜群环”代数和表示论。

H.Marubayashi: "On semi-hereditary orders integral over commutative valuation rings"Journal of Algebra. (to appear).

H.Marubayashi：“论交换估价环上的半遗传阶积分”代数杂志。

丸林英俊: "Dubrovin Valuation Properties of Skew Group Rings" Communications in Algebra. 26・1. 293-307 (1998)

Hidetoshi Marubayashi：“斜群环的杜布罗文估价性质”代数通讯 26・1 (1998)。