{"title":"交换环的湮灭理想图的推广","authors":"Mahtab Koohi Kerahroodi, Fatemeh Nabaei","doi":"10.4134/CKMS.C200006","DOIUrl":null,"url":null,"abstract":"Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, AG(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n,m ∈ N such that InJm = (0) with In, Jm 6= (0). First, we differentiate when AG(R) and AG(R) coincide. Then, we have characterized the diameter and the girth of AG(R) when R is a finite direct products of rings. Moreover, we show that AG(R) contains a cycle, if AG(R) 6= AG(R).","PeriodicalId":45637,"journal":{"name":"Communications of the Korean Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"AN EXTENSION OF ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS\",\"authors\":\"Mahtab Koohi Kerahroodi, Fatemeh Nabaei\",\"doi\":\"10.4134/CKMS.C200006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, AG(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n,m ∈ N such that InJm = (0) with In, Jm 6= (0). First, we differentiate when AG(R) and AG(R) coincide. Then, we have characterized the diameter and the girth of AG(R) when R is a finite direct products of rings. Moreover, we show that AG(R) contains a cycle, if AG(R) 6= AG(R).\",\"PeriodicalId\":45637,\"journal\":{\"name\":\"Communications of the Korean Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2020-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications of the Korean Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4134/CKMS.C200006\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications of the Korean Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4134/CKMS.C200006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设R是一个有单位的交换环。R的湮灭理想图AG(R)的扩展是顶点为R的非零湮灭理想且两个不同的顶点I和J相邻的图，当且仅当n,m∈n使得InJm =(0)与In, Jm 6=(0)。首先，我们区分AG(R)与AG(R)重合的情况。然后，我们刻画了当R是环的有限直积时AG(R)的直径和周长。此外，我们证明了AG(R)包含一个环，如果AG(R) 6= AG(R)。
AN EXTENSION OF ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS
Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, AG(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n,m ∈ N such that InJm = (0) with In, Jm 6= (0). First, we differentiate when AG(R) and AG(R) coincide. Then, we have characterized the diameter and the girth of AG(R) when R is a finite direct products of rings. Moreover, we show that AG(R) contains a cycle, if AG(R) 6= AG(R).
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This journal endeavors to publish significant research and survey of broad interests in pure and applied mathematics. One volume is published each year, and each volume consists of four issues (January, April, July, October).
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