环的相关强clean性

Since W.K.Nicholson introduced the notion of strongly clean rings in 1999, many algebraists have studied these rings with different view. Unitl now, there are still many meaningful unsolved questions. Herein we introduce three classes of special strongly clean rings: uniquely exchange rings, strongly J-clean rings and quasipolar rings. Using the theories, the way and techniques in ring theory, we study the relationship between these rings and strongly pi-rings, unit regular rings, uniquely (strongly) clean rings etc. With the view of module cancellation, the exchange property , continuous and discrete porperty, we will obtain the properties and characterizations of these three rings. We will construct many examples by studying some strong cleaness of incidence rings, power polynomial rings and extensions of rings. At last, we hope that some open questions will be solved (e.g., does every strongly clean rings have stable range 1? Is every unit regular ring strongly clean?), and the theory of involution ring, Banach algebra, C*-algebra will be riched. The results obtained in this work will be used in general inverse theory and algebraist K-theory.

自1999年W.K.Nicholson引入强clean环概念以来,广大代数学者从不同角度展开研究.但到目前为止,许多有意义的问题仍未解决.为此,我们引进三类特殊的强clean环: 惟一exchange环,强J-clean环和拟polar环. 借鉴环论中已有的理论、方法和技巧，研究它们与强 pi-正则,幺正则环, 惟一(强)clean环等重要环类的内在联系; 从模消去性,可替代性,连续离散性等角度入手,得到这些环的等价刻画和性质;通过研究incidence环,幂级数环和平凡扩张等环扩张的某些强clean性,构造更多例子.最终解决强clean环是否有稳定度1, 幺正则环是否强clean等公开问题,丰富involution环,Banach代数, C*-代数等代数结构的相关理论,并把所得结果应用到广义逆理论和代数K理论.

广大代数学者从不同角度研究强clean环，但仍有许多有意义的问题值得研究。本项目引进了三类特殊的强clean 环: 惟一exchange 环,强n-rad clean环和拟polar环. 研究的主要内容包括：（1）引入具有稳定度一的特殊强clean环：强n-rad clean环，并研究它的性质；（2）探讨了交换局部环上的2×2阶矩阵何时为强2-rad clean元, 给出了通过求解特征多项式来判别2×2阶矩阵是否为强2-rad clean元的方法.（3）研究了惟一clean元与惟一exchang元的关系，给出了Zariski topology下交换环的惟一clean元的刻画。项目申请者已根据上述研究的主要内容分别撰写了三篇论文，目前正在审稿中。