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

点击翻译按钮获取中文摘要

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.