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The trace graph of the matrix ring over a finite commutative ring

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Abstract

Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\), which we call the trace graph of the matrix ring \({M_n(R)}\), such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.

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References

  1. Akgünes, N.: Analyzing special parameters over zero-divisor graphs. AIP Conference Proceedings 1479, 390–393 (2012)

    Article  Google Scholar 

  2. Amini, A., Amini, B., Momtahan, E., Shirdareh, M.H.: Haghighi, On a graph of ideals. Acta Math. Hungar. 134, 369–384 (2012)

    Article  MathSciNet  Google Scholar 

  3. Anderson, D.F.: On the diameter and girth of a zero divisor graph. II. Houston J. Math. 34, 361–371 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Anderson, D.F., Badawi, A.: On the zero-divisor graph of a ring. Comm. Algebra 36, 3073–3092 (2008)

    Article  MathSciNet  Google Scholar 

  5. D. F. Anderson, M. Axtell and J. Stickles, Zero-divisor graphs in commutative rings, in: Commutative Algebra, Noetherian and Non-Noetherian Perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, I. Swanson, editors, Springer-Verlag (New York, 2010), pp. 23–45

    MATH  Google Scholar 

  6. Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)

    Article  MathSciNet  Google Scholar 

  7. Ashrafi, N., Maimani, H.R., Pournaki, M.R., Yassemi, S.: Unit graphs associated with rings. Comm. Algebra 38, 2851–2871 (2010)

    Article  MathSciNet  Google Scholar 

  8. Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)

    Article  MathSciNet  Google Scholar 

  9. Bozic, I.I., Petrovic, Z.: Zero-divisor graphs of matrices over commutative rings. Comm. Algebra 37, 1186–1192 (2009)

    Article  MathSciNet  Google Scholar 

  10. S. Kabbaj and A. Mimouni, Zero-divisor graphs of amalgamations, Math. Scand. (To appear)

  11. I. Kaplansky, Commutative Rings (rev. ed.), Univ. of Chicago Press (1974)

  12. T. Y. Lam, A First Course in Noncommutative Rings, Springer Science Business Media (Berlin–Heidelberg–New York, 1999)

  13. Li, A., Tucci, R.P.: Zero divisor graphs of upper triangular matrix rings. Comm. Algebra 41, 4622–4636 (2013)

    Article  MathSciNet  Google Scholar 

  14. Miguel, C.: Balanced zero-divisor graphs of matrix rings. Lobachevskii J. Math. 34, 137–141 (2013)

    Article  MathSciNet  Google Scholar 

  15. Mukwembi, S.: A note on diameter and the degree sequence of a graph. Appl. Math. Lett. 25, 175–178 (2012)

    Article  MathSciNet  Google Scholar 

  16. Mulay, S.B.: Cycles and symmetries of zero-divisors. Comm. Algebra 30, 3533–3558 (2002)

    Article  MathSciNet  Google Scholar 

  17. Okon, J.S.: Numbers of generators of ideals in a group ring of an elementary Abelian p-group. J. Algebra 224, 1–22 (2000)

    Article  MathSciNet  Google Scholar 

  18. Redmond, S.: The zero-divisor graph of a noncommutative ring. International J. Commutative Rings 1, 203–211 (2002)

    MATH  Google Scholar 

  19. Rush, D.: Rings with two-generated ideals. J. Pure Appl. Algebra 73, 257–275 (1991)

    Article  MathSciNet  Google Scholar 

  20. Rush, D.: Two-generated ideals and representations of abelian groups over valuation rings. J. Algebra 177, 77–101 (1995)

    Article  MathSciNet  Google Scholar 

  21. Smith, Z.O.: Planar zero-divisor graphs. Int. J. Commut. Rings 2, 177–188 (2003)

    MATH  Google Scholar 

  22. D. B. West, Introduction to Graph Theory, 2nd ed., Prentice Hall (2001)

  23. Wu, T., Yu, H., Lu, D.: The structure of finite local principal ideal rings. Comm. Algebra 40, 4727–4738 (2012)

    Article  MathSciNet  Google Scholar 

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Correspondence to F. A. A. Almahdi.

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Almahdi, F.A.A., Louartiti, K. & Tamekkante, M. The trace graph of the matrix ring over a finite commutative ring. Acta Math. Hungar. 156, 132–144 (2018). https://doi.org/10.1007/s10474-018-0815-x

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  • DOI: https://doi.org/10.1007/s10474-018-0815-x

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