Journal ArticleDOI

# A new graph associated to a commutative ring

26 May 2016-Discrete Mathematics, Algorithms and Applications (World Scientific Publishing Company)-Vol. 08, Iss: 02, pp 1650029
TL;DR: In this article, the authors consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) =(0).
Abstract: Let R be a commutative ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings R with at least two maximal ideals for which ΩR∗ are planar.
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TL;DR: The study of the essential ideal graph of a commutative ring with identity was initiated in this article, where the authors investigated its properties and showed that it is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal.
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TL;DR: The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-the theoretical properties of [/Formula]: see text.
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TL;DR: This paper characterize all Artinian rings [Formula: see text] for which the genus of [Form formula]: see text is zero or one.
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