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Some results on the annihilator graph of a commutative ring

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Abstract

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y), where for z ∈ R, ann R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ⩾ 1.

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Authors and Affiliations

  1. Department of Mathematics, University of Neyshabur, P.O.Box 91136–899, Neyshabur, Iran

    Mojgan Afkhami

  2. Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159–91775, Mashhad, Iran

    Kazem Khashyarmanesh & Zohreh Rajabi

Authors
  1. Mojgan Afkhami
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  2. Kazem Khashyarmanesh
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  3. Zohreh Rajabi
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Correspondence to Mojgan Afkhami.

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Afkhami, M., Khashyarmanesh, K. & Rajabi, Z. Some results on the annihilator graph of a commutative ring. Czech Math J 67, 151–169 (2017). https://doi.org/10.21136/CMJ.2017.0436-15

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  • DOI: https://doi.org/10.21136/CMJ.2017.0436-15

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