Divisor Graphs of a Commutative Ring

  • John D. LaGrangeEmail author
Part of the Trends in Mathematics book series (TM)


If x is an element of a commutative ring R then define the x-divisor graph \(\varGamma _x(R)\) to be the graph, whose vertices are the elements of \(d(x)=\{r\in R\) | \(rs=x\) for some \(s\in R\}\) such that two distinct vertices r and s are adjacent if and only if \(rs=x\). In this chapter, the components of \(\varGamma _x(R)\) are completely characterized when R is a von Neumann regular ring. Various other types of “divisor graphs” are considered as well. For example, if x is a nonzero element of an integral domain R with group of units U(R) then the compressed divisor graph \((\varGamma _E)_x^{d^\times }(R)\) associated with x is defined to be the graph, whose vertices are the associate-equivalence classes \(\overline{r}=rU(R)\) of elements \(r\in d(x)^\times =d(x)\setminus (xU(R)\cup U(R))\) such that two distinct vertices \(\overline{r}\) and \(\overline{s}\) are adjacent if and only if \(rs\in d(x)\). Alternatively, by letting M be the positive cone of the group of divisibility of R, every \((\varGamma _E)_x^{d^\times }(R)\) is a member of the class of graphs \(\varGamma _{\le x}(M)\) defined by picking an element x of a partially ordered commutative monoid M with least element equal to its identity 1, and letting the vertices of \(\varGamma _{\le x}(M)\) be the elements of \(\{m\in M\) | \(1<m<x\}\) such that two distinct vertices m and n are adjacent if and only if \(mn\le x\). Other aspects of the chapter include the exploration of graph-theoretic criteria that reveal when two elements of an integral domain are associates, and it is proved that R is a unique factorization domain if and only if \((\varGamma _E)_x^{d^\times }(R)\) is either null or finite with a dominant clique for every \(x\in R\setminus \{0\}\). Throughout, emphasis is placed on similarities with zero-divisor graphs. For example, it is proved that if R is von Neumann regular and G is a component of \(\varGamma _x(R)\) that contains a square root of x then \(G\cong \varGamma _0(\text {ann}_R(x))\) (in particular, if \(x=0\) then we have the tautology \(G\cong \varGamma _0(R)\)), and if x is a square-free element of a unique factorization domain then \((\varGamma _E)_x^{d^\times }(R)\) is isomorphic to a zero-divisor graph of a finite Boolean ring.


  1. 1.
    D.D. Anderson, M. Naseer, Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D.D. Anderson, D.F. Anderson, M. Zafrullah, Factorization in integral domains. J. Pure Appl. Algebra 69, 1–19 (1990)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.D. Anderson, D.F. Anderson, M. Zafrullah, Rings between \(D[X]\) and \(K[X]\). Houst. J. Math. 17, 109–129 (1991)Google Scholar
  4. 4.
    D.D. Anderson, J. Coykendall, L. Hill, M. Zafrullah, Monoid domain constructions of antimatter domains. Commun. Algebra 35, 3236–3241 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D.F. Anderson, A. Badawi, The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D.F. Anderson, A. Badawi, The zero-divisor graph of a commutative semigroup: a survey, in Groups, Modules, and Model Theory-Surveys and Recent Developments, ed. by M. Droste, L. Fuchs, B. Goldsmith, L. Strüngmann (Springer, Berlin, 2017), pp. 23–39CrossRefGoogle Scholar
  7. 7.
    D.F. Anderson, J.D. LaGrange, Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph. J. Pure Appl. Algebra 216, 1626–1636 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D.F. Anderson, J.D. LaGrange, Abian’s poset and the ordered monoid of annihilator classes in a reduced commutative ring. J. Algebra Appl. 13, 1450070(18 pp.) (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D.F. Anderson, J.D. LaGrange, Some remarks on the compressed zero-divisor graph. J. Algebra 447, 297–321 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D.F. Anderson, E.F. Lewis, A general theory of zero-divisor graphs over a commutative ring. Int. Electron. J. Algebra 20, 111–135 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D.F. Anderson, D. Weber, The zero-divisor graph of a commutative ring without identity. Int. Electron. J. Algebra 23, 176–202 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D.F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180, 221–241 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    D.F. Anderson, M.C. Axtell, J.A. Stickles Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspectives, ed. by M. Fontana, S.-E. Kabbaj, B. Olberding, I. Swanson (Springer, New York, 2011), pp. 23–45zbMATHGoogle Scholar
  15. 15.
    M. Axtell, J. Stickles, Irreducible divisor graphs in commutative rings with zero-divisors. Commun. Algebra 36, 1883–1893 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Axtell, M. Baeth, J. Stickles, Irreducible divisor graphs and factorization properties of domains. Commun. Algebra 39, 4148–4162 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Axtell, M. Baeth, J. Stickles, Survey article-graphical representations of factorizations in commutative rings. Rocky Mt. J. Math. 43, 1–36 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Axtell, M. Baeth, J. Stickles, Cut structures in zero-divisor graphs of commutative rings. J. Commut. Algebra 8, 143–171 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Badawi, On the annihilator graph of a commutative ring. Commun. Algebra 42, 108–121 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Badawi, On the dot product graph of a commutative ring. Commun. Algebra 43, 43–50 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    I. Beck, Coloring of commutative rings. J. Algebra 116, 208–226 (1988)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I. J. Algebra Appl. 10, 727–739 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    B. Bollobás, Modern Graph Theory (Springer, New York, 1998)CrossRefGoogle Scholar
  24. 24.
    P.M. Cohn, Bézout rings and their subrings. Math. Proc. Camb. Philos. Soc. 64, 251–264 (1968)CrossRefGoogle Scholar
  25. 25.
    J. Coykendall, J. Maney, Irreducible divisor graphs. Commun. Algebra 35, 885–895 (2007)MathSciNetCrossRefGoogle Scholar
  26. 26.
    J. Coykendall, D.E. Dobbs, B. Mullins, On integral domains with no atoms. Commun. Algebra 27, 5813–5831 (1999)MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, S. Spiroff, On zero divisor graphs, in Progress in Commutative Algebra II: Closures, Finiteness and Factorization, ed. by C. Francisco, L.C. Klinger, S.M. Sather-Wagstaff, J.C. Vassilev (de Gruyter, Berlin, 2012), pp. 241–299Google Scholar
  28. 28.
    F.R. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup. Semigroup Forum 65, 206–214 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    D.S. Dummit, R.M. Foote, Abstract Algebra, 3rd edn. (Wiley, New York, 2004)Google Scholar
  30. 30.
    J. Fröberg, An Introduction to Gröbner Bases (Wiley, New York, 1997)Google Scholar
  31. 31.
    R. Gilmer, Commutative Semigroup Rings (The University of Chicago Press, Chicago, 1984)zbMATHGoogle Scholar
  32. 32.
    R. Halaš, M. Jukl, On Beck’s coloring of partially ordered sets. Discret. Math. 309, 4584–4589 (2009)CrossRefGoogle Scholar
  33. 33.
    E. Hashemi, M. Abdi, A. Alhevaz, On the diameter of the compressed zero-divisor graph. Commun. Algebra 45, 4855–4864 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    V. Joshi, S. Sarode, Beck’s conjecture and multiplicative lattices. Discret. Math. 338, 93–98 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    C.F. Kimball, J.D. LaGrange, The idempotent-divisor graphs of a commutative ring. Commun. Algebra 46, 3899–3912 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    J.D. LaGrange, The x-divisor pseudographs of a commutative groupoid. Int. Electron. J. Algebra 22, 62–77 (2017)Google Scholar
  37. 37.
    J. Lambek, Lectures on Rings and Modules (Blaisdell Publishing Company, Waltham, 1966)zbMATHGoogle Scholar
  38. 38.
    D. Lu, T. Wu, The zero-divisor graphs of partially ordered sets and an application to semigroups. Graph Comb. 26, 793–804 (2010)CrossRefGoogle Scholar
  39. 39.
    X. Ma, D. Wang, J. Zhou, Automorphisms of the zero-divisor graph over \(2\times 2\) matrices. J. Korean Math. Soc. 53, 519–532 (2016)Google Scholar
  40. 40.
    J. Maney, Irreducible divisor graphs II. Commun. Algebra 36, 3496–3513 (2008)MathSciNetCrossRefGoogle Scholar
  41. 41.
    C.P. Mooney, Generalized irreducible divisor graphs. Commun. Algebra 42, 4366–4375 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    S.B. Mulay, Cycles and symmetries of zero-divisors. Commun. Algebra 30, 3533–3558 (2002)MathSciNetCrossRefGoogle Scholar
  43. 43.
    S.P. Redmond, An ideal based zero-divisor graph of a commutative ring. Commun. Algebra 31, 4425–4443 (2003)MathSciNetCrossRefGoogle Scholar
  44. 44.
    S. Spiroff, C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors. Commun. Algebra 39, 2338–2348 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Zaks, Half-factorial domains. Bull. Am. Math. Soc. 82, 721–724 (1976)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Lindsey Wilson CollegeColumbiaUSA

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