Synthetic Study of Prufer Orders and Semi-hereditary Orders

在2001-2004年间，我们研究了以下四个与非交换赋值环有关的问题：1.利用素段的概念给出了Dubrovin赋值环是完全有界的充要条件；2.利用局部化方法确定了Dubrovin赋值环的Martindale商环；3.研究了Ore扩张的非交换赋值环，得到了几个重要的非交换赋值环；4.设G是全序群，V是全赋值环.用分次理论方法研究了交叉积代数V_G的非交换赋值环。

For 2001-2004, we have studied the following four projects which are all concerned with non-commutative valuation rings :1.We gave a necessary and sufficient conditions for a Dubrovin valuation ring to be fully bounded by using the concept of the prime segments.2.Martindale quotient rings of Dubrovin valuation rings are determined in terms of the localizations.3.We have studied non-commutative valuation rings of Ore extensions and found out several important non-commutative valuation rings of Ore extensions.4.Let G be a totally ordered group and let V be a total valuation ring. We have studied non-commutative valuation rings of the crossed product algebra V G by the graded theoretical methods.