{"title":"交换Gelfand环上","authors":"A. R. Aliabad, M. Badie, S. Nazari","doi":"10.30495/JME.V0I0.1866","DOIUrl":null,"url":null,"abstract":"By studying and using the quasi-pure part concept, we improve some statements and show that some assumptions in some articles are superfluous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m(sum_{ α \\in A} I_ α) = sum_{ α \\in A} m ( I_ α ), for each family { I_ α}_{ α \\in A} of ideals of R , in addition if R is semiprimitive and Max( R ) ⊆ Y ⊆ Spec( R ), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec( R ) is regular. It has been shown that if R is a Gelfand ring, then Max( R ) is a quotient of Spec( R ), and sometimes h M ( a )’s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z Max( C ( X )) = { h_ M ( f ) : f \\in C ( X ) } if and only if { h_ M ( f ) : f \\in C ( X )} is closed under countable intersection if and only if X is pseudocompact.","PeriodicalId":43745,"journal":{"name":"Journal of Mathematical Extension","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2021-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On commutative Gelfand rings\",\"authors\":\"A. R. Aliabad, M. Badie, S. Nazari\",\"doi\":\"10.30495/JME.V0I0.1866\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"By studying and using the quasi-pure part concept, we improve some statements and show that some assumptions in some articles are superfluous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m(sum_{ α \\\\in A} I_ α) = sum_{ α \\\\in A} m ( I_ α ), for each family { I_ α}_{ α \\\\in A} of ideals of R , in addition if R is semiprimitive and Max( R ) ⊆ Y ⊆ Spec( R ), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec( R ) is regular. It has been shown that if R is a Gelfand ring, then Max( R ) is a quotient of Spec( R ), and sometimes h M ( a )’s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z Max( C ( X )) = { h_ M ( f ) : f \\\\in C ( X ) } if and only if { h_ M ( f ) : f \\\\in C ( X )} is closed under countable intersection if and only if X is pseudocompact.\",\"PeriodicalId\":43745,\"journal\":{\"name\":\"Journal of Mathematical Extension\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-10-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Extension\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30495/JME.V0I0.1866\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Extension","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30495/JME.V0I0.1866","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
通过对拟纯部分概念的研究和应用,我们改进了一些表述,证明了某些文章中的一些假设是多余的。给出了Gelfand环的一些特征。例如:证明R是格尔芬环当且仅当m(sum_{α \in A} I_ α) = sum_{α \in A} m(I_ α),对于R的理想的每一个族{I_ α} {α \in A},当且仅当R是半原元且Max(R),当Y是正态的,证明R是格尔芬环。证明了当R是约简环时,当且仅当Spec(R)是正则环时,R是von Neumann正则环。证明了如果R是一个Gelfand环,则Max(R)是Spec(R)的商,并且h M (a)有时表现为极大理想空间的零集。最后,证明了zmax (C (X)) = {h_ M (f): f \in C (X)}当且仅当{h_ M (f): f \in C (X)}是闭于可数交下的,当且仅当X是伪紧的。
By studying and using the quasi-pure part concept, we improve some statements and show that some assumptions in some articles are superfluous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m(sum_{ α \in A} I_ α) = sum_{ α \in A} m ( I_ α ), for each family { I_ α}_{ α \in A} of ideals of R , in addition if R is semiprimitive and Max( R ) ⊆ Y ⊆ Spec( R ), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec( R ) is regular. It has been shown that if R is a Gelfand ring, then Max( R ) is a quotient of Spec( R ), and sometimes h M ( a )’s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z Max( C ( X )) = { h_ M ( f ) : f \in C ( X ) } if and only if { h_ M ( f ) : f \in C ( X )} is closed under countable intersection if and only if X is pseudocompact.