Document Type : Original Manuscript

Authors

Department of Mathematics, Imam Khomeini International University, P.O.Box 34149-16818, Qazvin, Iran.

Abstract

‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎The‎
‎annihilator graph of $M$‎, ‎denoted by $AG(M)$ is a simple undirected‎
‎graph associated to $M$ whose the set of vertices is‎
‎$Z_R(M) \setminus {\rm Ann}_R(M)$ and two distinct vertices $x$ and‎
‎$y$ are adjacent if and only if ${\rm Ann}_M(xy)\neq {\rm‎ ‎Ann}_M(x) \cup {\rm Ann}_M(y)$‎. ‎In this paper‎, ‎we study the‎
‎diameter and the girth of $AG(M)$ and we characterize all modules‎
‎whose annihilator graph is complete‎. ‎Furthermore‎, ‎we look for the‎
‎relationship between the annihilator graph of $M$ and its zero-divisor‎
‎graph‎.

Keywords

###### ##### References
1. D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.
2. D.F. Anderson and S.B. Mulay, On the diameter and girth of a zero-divisor graph, J. Algebra, 210 (2007), 543–550.

3. S. Babaei, Sh. Payrovi and E. Sengelen Sevim, On the annihilator submodules and the annihilator essential graph, Acta Math. Vietnam., 44 (2019), 905–914.

4. A. Badawi, On the 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc., 75 (2007), 417–429.

5. A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108–121.

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

7. M. Behboodi, Zero-divisor graph for modules over commutative rings, J. Commut. Algebra, 4 (2012), 175–197.

8. S. Dutta and C. Lanong, On annihilator graph of a finite commutative ring, Trans. Comb., 6 (2017), 1–11.

9. A.R. Naghipour, The zero-divisor graph of a module, J. Algebr. Syst., 4 (2017), 155–171.

10. M.J. Nikmehr, R. Nikandish and M. Bakhtyiari, More on the annihilator graph of a commutative ring, Hokkaido Math. J., 46 (2017), 1–12.

11. K. Nozari and Sh. Payrovi, A generalization of zero-divisor graph for modules, Publ. Inst. Math., 106 (2019), 39–46·

12. Sh. Payrovi and S. Babaei, On 2-absorbing submodules, Algebra Colloq., 19 (2012), 913–920.
13. Sh. Payrovi and S. Babaei, On the 2-absorbing submodules, Iran. J. Math. Sci. Inform., 10 (2015), 131–137.
14. Sh. Payrovi and S. Babaei, On the compressed annihilator graph of a commutative ring, Indian J. Pure and Appl. Math., 49 (2018), 177–186.

15. R.Y. Sharp, Steps in Commutative Algebra, Cambridge University Press, Cambridge, 2000.