# Divisor Graphs of a Commutative Ring

• John D. LaGrange
Chapter
Part of the Trends in Mathematics book series (TM)

## 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. 1.
D.D. Anderson, M. Naseer, Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993)
2. 2.
D.D. Anderson, D.F. Anderson, M. Zafrullah, Factorization in integral domains. J. Pure Appl. Algebra 69, 1–19 (1990)
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)
5. 5.
D.F. Anderson, A. Badawi, The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)
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. 23–39
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)
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)
9. 9.
D.F. Anderson, J.D. LaGrange, Some remarks on the compressed zero-divisor graph. J. Algebra 447, 297–321 (2016)
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)
11. 11.
D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
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)
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)
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. 23–45
15. 15.
M. Axtell, J. Stickles, Irreducible divisor graphs in commutative rings with zero-divisors. Commun. Algebra 36, 1883–1893 (2008)
16. 16.
M. Axtell, M. Baeth, J. Stickles, Irreducible divisor graphs and factorization properties of domains. Commun. Algebra 39, 4148–4162 (2011)
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)
18. 18.
M. Axtell, M. Baeth, J. Stickles, Cut structures in zero-divisor graphs of commutative rings. J. Commut. Algebra 8, 143–171 (2016)
19. 19.
A. Badawi, On the annihilator graph of a commutative ring. Commun. Algebra 42, 108–121 (2014)
20. 20.
A. Badawi, On the dot product graph of a commutative ring. Commun. Algebra 43, 43–50 (2015)
21. 21.
I. Beck, Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
22. 22.
M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I. J. Algebra Appl. 10, 727–739 (2011)
23. 23.
B. Bollobás, Modern Graph Theory (Springer, New York, 1998)
24. 24.
P.M. Cohn, Bézout rings and their subrings. Math. Proc. Camb. Philos. Soc. 64, 251–264 (1968)
25. 25.
J. Coykendall, J. Maney, Irreducible divisor graphs. Commun. Algebra 35, 885–895 (2007)
26. 26.
J. Coykendall, D.E. Dobbs, B. Mullins, On integral domains with no atoms. Commun. Algebra 27, 5813–5831 (1999)
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)
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)
32. 32.
R. Halaš, M. Jukl, On Beck’s coloring of partially ordered sets. Discret. Math. 309, 4584–4589 (2009)
33. 33.
E. Hashemi, M. Abdi, A. Alhevaz, On the diameter of the compressed zero-divisor graph. Commun. Algebra 45, 4855–4864 (2017)
34. 34.
V. Joshi, S. Sarode, Beck’s conjecture and multiplicative lattices. Discret. Math. 338, 93–98 (2015)
35. 35.
C.F. Kimball, J.D. LaGrange, The idempotent-divisor graphs of a commutative ring. Commun. Algebra 46, 3899–3912 (2018)
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)
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)
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)
41. 41.
C.P. Mooney, Generalized irreducible divisor graphs. Commun. Algebra 42, 4366–4375 (2014)
42. 42.
S.B. Mulay, Cycles and symmetries of zero-divisors. Commun. Algebra 30, 3533–3558 (2002)
43. 43.
S.P. Redmond, An ideal based zero-divisor graph of a commutative ring. Commun. Algebra 31, 4425–4443 (2003)
44. 44.
S. Spiroff, C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors. Commun. Algebra 39, 2338–2348 (2011)
45. 45.
A. Zaks, Half-factorial domains. Bull. Am. Math. Soc. 82, 721–724 (1976)