论文标题
关于应用程序的纯粹理想
On purely-prime ideals with applications
论文作者
论文摘要
在本文中，获得了通勤环（纯频谱）的纯粹主机理想的新代数和拓扑结果。特别是，获得了Grothendieck Type定理，该定理指出，环的同性恋者与其纯光谱的clopens之间存在规范对应关系。还证明，给定的环是gelfand环，如果它的最大光谱配备了诱导的Zariski拓扑结构，则与其纯光谱同构。然后作为应用，推断一个环为零的IFF尺寸为零，纯谱是同构的。双重表明，给定的环是MP环的减小IFF，其最小频谱配备了诱导的扁平拓扑，并且其纯光谱是相同的。最后，引入了新的noetherian环的新概念，并证明了科恩型定理。
In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck type theorem is obtained which states that there is a canonical correspondence between the idempotents of a ring and the clopens of its pure spectrum. It is also proved that a given ring is a Gelfand ring iff its maximal spectrum equipped with the induced Zariski topology is homeomorphic to its pure spectrum. Then as an application, it is deduced that a ring is zero dimensional iff its prime spectrum and pure spectrum are isomorphic. Dually, it is shown that a given ring is a reduced mp-ring iff its minimal spectrum equipped with the induced flat topology and its pure spectrum are the same. Finally, the new notion of semi-Noetherian ring is introduced and Cohen type theorem is proved.
