Toroidal zero-divisor graphs of decomposable commutative rings without identity

Abstract

Let R be a commutative ring without identity. The zero-divisor graph of R,  denoted by \(\varGamma (R),\) is a graph with vertex set \(Z(R){{\setminus }} \{0\},\) which is the set of all non-zero zero-divisor elements of R and two vertices x and y are adjacent if and only if \(xy=0.\) In this paper, we characterize (up to isomorphism) all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal.

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Acknowledgement

The authors express their sincere thanks for the anonymous referee for careful reading and suggestions which improved the presentation of the paper.

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Correspondence to T. Tamizh Chelvam.

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Kalaimurugan, G., Vignesh, P. & Tamizh Chelvam, T. Toroidal zero-divisor graphs of decomposable commutative rings without identity. Bol. Soc. Mat. Mex. 26, 807–829 (2020). https://doi.org/10.1007/s40590-020-00282-3

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Keywords

  • Commutative rings
  • Nilpotent rings
  • Decomposable rings
  • Zero-divisor graph
  • Genus

Mathematics Subject Classification

  • 05C10
  • 05C25
  • 13M05