Czechoslovak Mathematical Journal, ( ISI ), Volume (67), No (1), Year (2017-2) , Pages (151-169)

Title : ( Some results on the annihilator graph of~a~commutative ring )

Authors: M. Afkhami , Kazem Khashyarmanesh , zohreh rajabi ,

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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

Keywords

, annihilator graph, zero-divisor graph, outerplanar, ring-graph, cut-vertex, clique number, weakly perfect, chromatic number, polynomial ring, ring of fractions
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@article{paperid:1067048,
author = {M. Afkhami and Khashyarmanesh, Kazem and Rajabi, Zohreh},
title = {Some results on the annihilator graph of~a~commutative ring},
journal = {Czechoslovak Mathematical Journal},
year = {2017},
volume = {67},
number = {1},
month = {February},
issn = {1572-9141},
pages = {151--169},
numpages = {18},
keywords = {annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions},
}

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%0 Journal Article
%T Some results on the annihilator graph of~a~commutative ring
%A M. Afkhami
%A Khashyarmanesh, Kazem
%A Rajabi, Zohreh
%J Czechoslovak Mathematical Journal
%@ 1572-9141
%D 2017

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