Advances in Commutative Algebra pp 217-244 | Cite as

# Divisor Graphs of a Commutative Ring

## Abstract

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.

## References

- 1.D.D. Anderson, M. Naseer, Beck’s coloring of a commutative ring. J. Algebra
**159**, 500–514 (1993)MathSciNetCrossRefGoogle Scholar - 2.D.D. Anderson, D.F. Anderson, M. Zafrullah, Factorization in integral domains. J. Pure Appl. Algebra
**69**, 1–19 (1990)MathSciNetCrossRefGoogle Scholar - 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.D.D. Anderson, J. Coykendall, L. Hill, M. Zafrullah, Monoid domain constructions of antimatter domains. Commun. Algebra
**35**, 3236–3241 (2007)MathSciNetCrossRefGoogle Scholar - 5.D.F. Anderson, A. Badawi, The total graph of a commutative ring. J. Algebra
**320**, 2706–2719 (2008)MathSciNetCrossRefGoogle Scholar - 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.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.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.D.F. Anderson, J.D. LaGrange, Some remarks on the compressed zero-divisor graph. J. Algebra
**447**, 297–321 (2016)MathSciNetCrossRefGoogle Scholar - 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.D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra
**217**, 434–447 (1999)MathSciNetCrossRefGoogle Scholar - 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.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.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.M. Axtell, J. Stickles, Irreducible divisor graphs in commutative rings with zero-divisors. Commun. Algebra
**36**, 1883–1893 (2008)MathSciNetCrossRefGoogle Scholar - 16.M. Axtell, M. Baeth, J. Stickles, Irreducible divisor graphs and factorization properties of domains. Commun. Algebra
**39**, 4148–4162 (2011)MathSciNetCrossRefGoogle Scholar - 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.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.A. Badawi, On the annihilator graph of a commutative ring. Commun. Algebra
**42**, 108–121 (2014)MathSciNetCrossRefGoogle Scholar - 20.A. Badawi, On the dot product graph of a commutative ring. Commun. Algebra
**43**, 43–50 (2015)MathSciNetCrossRefGoogle Scholar - 21.I. Beck, Coloring of commutative rings. J. Algebra
**116**, 208–226 (1988)MathSciNetCrossRefGoogle Scholar - 22.M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I. J. Algebra Appl.
**10**, 727–739 (2011)MathSciNetCrossRefGoogle Scholar - 23.B. Bollobás,
*Modern Graph Theory*(Springer, New York, 1998)CrossRefGoogle Scholar - 24.P.M. Cohn, Bézout rings and their subrings. Math. Proc. Camb. Philos. Soc.
**64**, 251–264 (1968)CrossRefGoogle Scholar - 25.J. Coykendall, J. Maney, Irreducible divisor graphs. Commun. Algebra
**35**, 885–895 (2007)MathSciNetCrossRefGoogle Scholar - 26.J. Coykendall, D.E. Dobbs, B. Mullins, On integral domains with no atoms. Commun. Algebra
**27**, 5813–5831 (1999)MathSciNetCrossRefGoogle Scholar - 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.F.R. DeMeyer, T. McKenzie, K. Schneider, The zero-divisor graph of a commutative semigroup. Semigroup Forum
**65**, 206–214 (2002)MathSciNetCrossRefGoogle Scholar - 29.D.S. Dummit, R.M. Foote,
*Abstract Algebra*, 3rd edn. (Wiley, New York, 2004)Google Scholar - 30.J. Fröberg,
*An Introduction to Gröbner Bases*(Wiley, New York, 1997)Google Scholar - 31.R. Gilmer,
*Commutative Semigroup Rings*(The University of Chicago Press, Chicago, 1984)zbMATHGoogle Scholar - 32.R. Halaš, M. Jukl, On Beck’s coloring of partially ordered sets. Discret. Math.
**309**, 4584–4589 (2009)CrossRefGoogle Scholar - 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.V. Joshi, S. Sarode, Beck’s conjecture and multiplicative lattices. Discret. Math.
**338**, 93–98 (2015)MathSciNetCrossRefGoogle Scholar - 35.C.F. Kimball, J.D. LaGrange, The idempotent-divisor graphs of a commutative ring. Commun. Algebra
**46**, 3899–3912 (2018)MathSciNetCrossRefGoogle Scholar - 36.J.D. LaGrange, The
*x*-divisor pseudographs of a commutative groupoid. Int. Electron. J. Algebra**22**, 62–77 (2017)Google Scholar - 37.J. Lambek,
*Lectures on Rings and Modules*(Blaisdell Publishing Company, Waltham, 1966)zbMATHGoogle Scholar - 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.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.J. Maney, Irreducible divisor graphs II. Commun. Algebra
**36**, 3496–3513 (2008)MathSciNetCrossRefGoogle Scholar - 41.C.P. Mooney, Generalized irreducible divisor graphs. Commun. Algebra
**42**, 4366–4375 (2014)MathSciNetCrossRefGoogle Scholar - 42.S.B. Mulay, Cycles and symmetries of zero-divisors. Commun. Algebra
**30**, 3533–3558 (2002)MathSciNetCrossRefGoogle Scholar - 43.S.P. Redmond, An ideal based zero-divisor graph of a commutative ring. Commun. Algebra
**31**, 4425–4443 (2003)MathSciNetCrossRefGoogle Scholar - 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.A. Zaks, Half-factorial domains. Bull. Am. Math. Soc.
**82**, 721–724 (1976)MathSciNetCrossRefGoogle Scholar