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

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。

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.