The weakly zero-divisor graph of a commutative ring

• M.J. Nikmehr ^[1] ; A. Azadi ^[1] ; R. Nikandish ^[2]
  1. [1] University of Technology, Teherán, Irán
  2. [2] University of Technology, Dezful, Irán
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 62, Nº. 1, 2021, págs. 105-116
• Idioma: inglés
• DOI: 10.33044/revuma.1677
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• Resumen

  • Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph WΓ(R) with vertex set Z(R) ∗, and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0. It follows that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of WΓ(R) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of WΓ(R) is studied.