2017/07/15 by M. J. Nikmehr, Mehdi Hosseini, Nikmehr, M. J. +2
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Annihilator #Combinatorics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Commutative property #Commutative ring #Discrete mathematics #FOS: Mathematics #Graph #Ideal (ethics) #Mathematics #Philosophy #Pure mathematics #Rings, Modules, and Algebras #Simple graph #Vertex (graph theory) #math.AC #math.CO
paper · pdf · doi:10.48550/arxiv.1707.04697
published in arXiv (Cornell University) (Cornell University)
arxiv created 2017/07/15 · openalex publication_date 2017/07/15 · arxiv updated 2017/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let R be a commutative ring with identity and \Bbb A (R) be the set of ideals of R with non-zero annihilator. The annihilator-ideal graph of R, denoted by AI (R) , is a simple graph with the vertex set \Bbb A(R)∗ := \Bbb A (R) ∖\lbrace (0) \rbrace , and two distinct vertices I and J are adjacent if and only if Ann R (IJ) ≠ Ann R (I) ∪ Ann R (J). In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph \Bbb A \Bbb G (R) (a well-known graph with the same vertices and two distinct vertices I,J are adjacent if and only if IJ=0) associated with R. All rings whose AI(R) ≠ \Bbb A \Bbb G (R) and gr (AI(R)) =4 are characterized. Among other results, we obtain necessary and sufficient conditions under which AI (R) is a star graph.