【作者】 庞佳；

【导师】 王芳贵；

【作者基本信息】 四川师范大学 ， 基础数学， 2009， 硕士

【摘要】 近年来关于PVMR的研究见诸于不少文献,一直受到人们的关注.本文运用伪局部化方法与一般交换环上w-算子刻画PVMR.第一章首先引入伪局部化R[S]与S-可约,并探讨了R[S]的一些基本性质.然后,将一般交换环上的w-算子与伪局部化R[S]相结合,得到模的伪局部整体原理.第二章首先介绍了Manis赋值理论,定义了伪赋值环,即P是R的唯一正则极大理想且(R,P)是赋值对.并结合Manis赋值理论与伪局部整体原理,得到PVMR的等价刻画,即R是PVMR当且仅当对R的任何的极大w-理想Q, (R[Q],[Q]R[Q])是Manis赋值环;这也等价于对于R的任何的正则极大w-理想P, (R[P],[P]R[P])是伪赋值环.其次,采用传统的理想理论的研究方法,结合w-算子得到PVMR的另一等价刻画,即R是PVMR当且仅当R的每个有限型的正则理想是w-平坦理想(w-投射理想);若R是弱DW环,则R是PVMR当且仅当R是Pru|¨fer环.更多还原

【Abstract】 PVMRs have received a good deal of attention continuously in a number ofliterature in recent decade. In this thesis, PVMRs are characterized by utiliz-ing pseudo-localization principle and w-operations. In chapter 1, the concept ofpseudo-localization R[S] and S-cancellation are introduced. Then, we obtain someproperties of pseudo-localization R[S]. Moreover, the pseudo-local to global prin-ciple is received by using pseudo-localization R[S] and w-operations. In chapter2, Manis valuation theory is presented. Pseudo-valuation ring R is defined, whichhave a unique regular maximal ideal P and (R,P) is a valuation pair. On theone hand, by using pseudo-localization principle and w-operations, it is shownthat R is a PVMR if and only if (R[Q],[Q]R[Q]) is a Manis valuation ring for anymaximal w-ideal Q of R; which is equivalent to that (R[P],[P]R[P]) is a pseudo-valuation ring for any regular maximal w-ideal P of R. On the other hand, byusing traditional ideal theory methods, we proved that R is a PVMR if and onlyif every finite type ideal of R is w-projective(w-flat). If R is a weak DW ring,then R is a PVMR if and only if R is a Pru|¨fer ring.更多还原