{"title":"坐标系上连续整数值函数环的理想","authors":"T. Dube, O. Ighedo, Batsile Tlharesakgosi","doi":"10.36045/j.bbms.210412","DOIUrl":null,"url":null,"abstract":"Let L be a zero-dimensional frame and Z L be the ring of integer-valued continuous functions on L . We associate with each sublocale of ζL , the Banaschewski compactification of L , an ideal of Z L , and show the behaviour of these types of ideals. The socle of Z L is shown to be always the zero ideal, in contrast with the fact that the socle of the ring R L of continuous real-valued functions on L is not necessarily the zero ideal. The ring Z L has been shown by B. Banaschewski to be (isomorphic to) a subring of R L , so that the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of R L to Z L is the ideal of Z L associated with the join (in the coframe of sublocales of ζL ) of all nowhere dense sublocales of ζL . It also appears in other guises.","PeriodicalId":55309,"journal":{"name":"Bulletin of the Belgian Mathematical Society-Simon Stevin","volume":"222 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On ideals of rings of continuous integer-valued functions on a frame\",\"authors\":\"T. Dube, O. Ighedo, Batsile Tlharesakgosi\",\"doi\":\"10.36045/j.bbms.210412\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let L be a zero-dimensional frame and Z L be the ring of integer-valued continuous functions on L . We associate with each sublocale of ζL , the Banaschewski compactification of L , an ideal of Z L , and show the behaviour of these types of ideals. The socle of Z L is shown to be always the zero ideal, in contrast with the fact that the socle of the ring R L of continuous real-valued functions on L is not necessarily the zero ideal. The ring Z L has been shown by B. Banaschewski to be (isomorphic to) a subring of R L , so that the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of R L to Z L is the ideal of Z L associated with the join (in the coframe of sublocales of ζL ) of all nowhere dense sublocales of ζL . It also appears in other guises.\",\"PeriodicalId\":55309,\"journal\":{\"name\":\"Bulletin of the Belgian Mathematical Society-Simon Stevin\",\"volume\":\"222 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Belgian Mathematical Society-Simon Stevin\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.36045/j.bbms.210412\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Belgian Mathematical Society-Simon Stevin","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.36045/j.bbms.210412","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
On ideals of rings of continuous integer-valued functions on a frame
Let L be a zero-dimensional frame and Z L be the ring of integer-valued continuous functions on L . We associate with each sublocale of ζL , the Banaschewski compactification of L , an ideal of Z L , and show the behaviour of these types of ideals. The socle of Z L is shown to be always the zero ideal, in contrast with the fact that the socle of the ring R L of continuous real-valued functions on L is not necessarily the zero ideal. The ring Z L has been shown by B. Banaschewski to be (isomorphic to) a subring of R L , so that the ideals of the larger ring can be contracted to the smaller one. We show that the contraction of the socle of R L to Z L is the ideal of Z L associated with the join (in the coframe of sublocales of ζL ) of all nowhere dense sublocales of ζL . It also appears in other guises.
期刊介绍:
The Bulletin of the Belgian Mathematical Society - Simon Stevin (BBMS) is a peer-reviewed journal devoted to recent developments in all areas in pure and applied mathematics. It is published as one yearly volume, containing five issues.
The main focus lies on high level original research papers. They should aim to a broader mathematical audience in the sense that a well-written introduction is attractive to mathematicians outside the circle of experts in the subject, bringing motivation, background information, history and philosophy. The content has to be substantial enough: short one-small-result papers will not be taken into account in general, unless there are some particular arguments motivating publication, like an original point of view, a new short proof of a famous result etc.
The BBMS also publishes expository papers that bring the state of the art of a current mainstream topic in mathematics. Here it is even more important that at leat a substantial part of the paper is accessible to a broader audience of mathematicians.
The BBMS publishes papers in English, Dutch, French and German. All papers should have an abstract in English.