{"title":"交换环的素数理想图","authors":"H. M. Salih, Asaad A. Jund","doi":"10.19184/ijc.2022.6.1.2","DOIUrl":null,"url":null,"abstract":"Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that ΓP is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of ΓP.","PeriodicalId":13506,"journal":{"name":"Indonesian Journal of Combinatorics","volume":"110 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Prime ideal graphs of commutative rings\",\"authors\":\"H. M. Salih, Asaad A. Jund\",\"doi\":\"10.19184/ijc.2022.6.1.2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that ΓP is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of ΓP.\",\"PeriodicalId\":13506,\"journal\":{\"name\":\"Indonesian Journal of Combinatorics\",\"volume\":\"110 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Indonesian Journal of Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.19184/ijc.2022.6.1.2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indonesian Journal of Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.19184/ijc.2022.6.1.2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that ΓP is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of ΓP.
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