论文标题
谐波环和戒指的结构结果
Structural results on harmonic rings and lessened rings
论文作者
论文摘要
在本文中，使用代数和拓扑方法的组合来获得有关谐波环的新结构结果。特别是，如果表明，如果Gelfand Ring $ a $ modulo其Jacobson Radical是零维戒指，那么$ A $就是一个干净的环。还证明，对于给定的Gelfand Ring $ a $，然后，回缩地图规格$（a）\ rightarrow {\ rm max}（a）$在且仅当$ a $ a $ a $ modulo的jacobson激进分子是零二光环时是连续的。双重证明，对于给定的MP-RING $ A $，则回缩映射规格$（a）\ rightArrow {\ rm min}（a）$是Zariski的连续，并且仅当$ {\ rm min}（a）$是Zariski compact。给出了零维环，MP环和Gelfand环的新标准。引入和研究了新的减少环的概念，该概念概括了“还原环”概念。特别是获得技术结果，该技术结果指出，当且仅当每个因素都是一个减少的环时，一个环家族的产物是一个减少的环。作为这种精神的另一个结果，还表征了局部减少的MP环的结构。最后，特征是当给定的环是田地，整体域，本地环和较小的准派环的有限产物时。
In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring $A$ modulo its Jacobson radical is a zero dimensional ring, then $A$ is a clean ring. It is also proved that, for a given Gelfand ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Max}(A)$ is flat continuous if and only if $A$ modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Min}(A)$ is Zariski continuous if and only if ${\rm Min}(A)$ is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes "reduced ring" notion. Specially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring when is a finite product of fields, integral domains, local rings, and lessened quasi-prime rings.
