A study of integrated orders and its applications

我们主要研究了以下五种不同类型环的代数结构。(1)设R为半单环上的偏斜多项式环。证明了R中有三种不同的中心幂等，进一步确定了R的素根的结构，证明了R的任何素理想要么是极大理想，要么是主理想，要么是幂等的。(2)设R是一个非交换赋值环上的Ore扩展。我们证明了R总是广义Bezout。进一步得到了R完全有界的一个充分必要条件。(3)对非交换赋值环上交叉积代数的素理想进行了分类。(4)对普鲁弗环的定域系统进行了分类。(5)在可交换Dedekind域上对歪斜多项式环上的素理想进行了分类，并确定了素因子环的结构。

We have mainly studied the algebraic structure of the following five different types of rings.(1) Let R be a partial skew polynomial ring over a semi-simple Artinain ring. We proved there are three different central idempotent in R. Furthermore we determined the structure of prime radical of R and proved that any prime ideals of R is either a maximal ideal and principal or idempotent.(2) Let R be an Ore extension over a non-commutative valuation ring. We proved that R is always a generalized Bezout. Further we obtained that a necessary and sufficient conditions for R to be fully bounded.(3) We classified the prime ideals of crossed product algebras over non-commutative valuation rings.(4) We classified the localizing system of Prufer rings.(5) We classified the prime ideals in skew polynomial rings over commutative Dedekind domain and determined the structure of the prime factor rings.