幂等元的提升、正交性和有限性的结构性结果
Structural results on lifting, orthogonality and finiteness of idempotents
摘要：
本文利用幂等元与闭序圈之间的典范对应，得到了提升幂等元的几个新结果。精确地确定了极大谱的Zebraki闭环，作为应用，刻画了模Jacobson根的提升幂等元.提升幂等元模任意理想也可以用与该理想相关的某些连通集来表征。作为应用，得到了提升理想与正则理想的和是提升理想。证明了提升幂等元在可数情形下保持正交性。刻画了环的任意态射的提升性质。作为另一个主要结果，证明了环$R$的幂等元个数是有限的当且仅当它是$2^κ$的形式，其中$κ$是Spec $(R)$的连通分支的基数.最后证明了零维环的本原幂等元与其素谱的孤立点是1-1对应的。这些结果推广或改进了文献中的几个重要结果。
英文摘要：
In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring $R$ is finite if and only if it is of the form $2^κ$ where $κ$ is the cardinal of the connected components of Spec$(R)$. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature.
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