Zero intersection graph of annihilator ideals of modules

Year 2025, Volume: 54 Issue: 6, 2182 - 2194, 30.12.2025

Seçil Çeken , Osama A. Naji , Ünsal Tekir , Suat Koç

https://doi.org/10.15672/hujms.1485903

Abstract

This paper aims to associate a new graph to nonzero unital modules over commutative rings. Let $R$ be a commutative ring having a nonzero identity and $M$ be a nonzero unital $R$-module. The zero intersection graph of annihilator ideals of $R$-module $M$, denoted by $\mathfrak{C}_{R}(M)$, is a simple (undirected) graph whose vertex set $M^{\star}=M-\{0\},\ $and two distinct vertices $m$ and $m^{\prime}$ are adjacent if $ann_{R}(m)\cap ann_{R}(m^{\prime})=(0).$\ We investigate the conditions under which $\mathfrak{C}_{R}(M)$ is a star graph, bipartite graph, complete graph, edgeless graph. Furthermore, we characterize certain classes of modules and rings such as torsion-free modules, torsion modules, semisimple modules, quasi-regular rings, and modules satisfying Property $T$ in terms of their graphical properties.

Keywords

property A , property T , torsion-free module , quasi regular ring , weak quasi regular module

References

• [1] S. Akbaria, H.A. Tavallaeeb and S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl. 10, 1-8, 2011.
• [2] D.D. Anderson, T. Arabaci, Ü. Tekir and S. Koç, On S-multiplication modules, Comm. Algebra 48(8), 3398-3407, 2020.
• [3] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320(7), 2706-2719, 2008.
• [4] D.D. Anderson and S. Chun, The set of torsion elements of a module, Comm. Algebra 42(4), 1835-1843, 2014.
• [5] D.D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl. 16(08), 1750143, 2017.
• [6] D.D. Anderson and S. Chun, McCoy modules and related modules over commutative rings, Comm. Algebra 45(6), 2593-2601, 2017.
• [7] F.W. Anderson and K.R. Fuller, Rings and categories of modules (Vol. 13), Springer Science & Business Media, 2012.
• [8] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42(1), 108-121, 2014.
• [9] A. Barnard, Multiplication modules, J. Algebra. 71(1), 174-178, 1981.
• [10] I. Beck, Coloring of commutative rings, J. Algebra 116(1), 208-226, 1988.
• [11] M. Behboodi, Zero divisor graphs for modules over commutative rings, J. Commut. Algebra 4(2), 175-197, 2012.
• [12] D. Bennis, J. Mikram and F. Taraza, On the extended zero divisor graph of commutative rings, Turk. J. Math. 40(2), 376-388, 2016.
• [13] I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309, 5381-5392, 2009.
• [14] G. Chartrand and P. Zhang, A First Course in Graph Theory, Dover Publications, New York, 2012.
• [15] A.Y. Darani, Notes on annihilator conditions in modules over commutative rings, An. Stinnt. Univ. Ovidius Constanta 18, 59-72, 2010.
• [16] R. Diestel, Graph Theory, Springer-Verlag, New York, 1997.
• [17] M. Evans, On commutative PP rings, Pacific J. Math. 41(3), 687-697, 1972.
• [18] Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra 16(4), 755-779, 1988.
• [19] Y. El-Khabchi, E.M. Bouba and S. Koç, On the global powerful alliance number of zero-divisor graphs of finite commutative rings, J. Algebra Appl. 24(3), 2550089, 2025.
• [20] J.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker Inc., 1988.
• [21] C. Jayaram and Ü. Tekir, von Neumann regular modules, Comm. Algebra 46(5), 2205-2217, 2018.
• [22] C. Jayaram, Ü. Tekir and S. Koç, On Quasi regular modules and trivial extension, Hacettepe J. Math. Stat. 50(1), 120-134, 2021.
• [23] C. Jayaram, Ü. Tekir and S. Koç, On Baer modules, Rev. Un. Mat. Argentina 63(1), 109-128, 2022
• [24] C. Jayaram, Ü. Tekir, S. Koç and S. Çeken, On normal modules, Comm. Algebra 51(4), 1479-1491, 2023.
• [25] M.L. Knox, R. Levy, W.W. McGovern and J. Shapiro, Generalizations of complemented rings with applications to rings of functions, J. Algebra Appl. 8(1), 17-40, 2009.
• [26] S. Koç, On Strongly $\pi$-regular Modules, Sakarya Uni. J. Science 24(4), 675-684, 2020.
• [27] M.D. Larsen and P.J. MacCarthy, Multiplicative theory of ideals, New York: Academic Press, 1971.
• [28] T.K. Lee and Y. Zhou, Reduced modules, Rings, modules, algebras and abelian groups 236, Lecture Notes in Pure and Appl. Mathematics, pp. 365-377, Marcel Dekker, New York, 2004.
• [29] R. Levy and J. Shapiro, The zero-divisor graph of von Neumann regular rings, Comm. Algebra 30(2), 745-750, 2002.
• [30] N. Mahdou and A.R. Hassani, On strong (A)-rings, Mediterr. J. Math. 9(2), 393-402, 2012.
• [31] P.M. Rad, S. Yassemi, S. Ghalandarzadeh and P. Safari, Diameter and girth of Torsion Graph, An. tiin. Univ. Ovidius Constanta 22(3), 127-136, 2014.
• [32] P.M. Rad, Planar Torsion Graph of Modules, Filomat 30(2), 367-372, 2016.
• [33] H. Mostafanasab and A.Y. Darani, 2-Irreducible and Strongly 2-Irreducible ideals of commutative rings, Miskolc Math. Notes 17(1), 441-455, 2016.
• [34] M. Nazim and N. Rehman, On the essential annihilating-ideal graph of commutative rings, Ars Math. Contemp. 22(3), (16 pages), 2022.
• [35] M. Nazim, N.U.Rehman and K. Selvakumar, On the genus of annihilator intersection graph of commutative rings, Alg. Struc. Appl. 11(1), 25-36, 2024.
• [36] M.J. Nikmehr and S. Khojasteh, On the nilpotent graph of a ring, Turk. J. Math. 37(4), 553-559, 2013.
• [37] S. Payrovi and S. Babaei, The compressed annihilator graph of a commutative ring, Indian J. Pure Appl. Math. 49(1), 177-186, 2018.
• [38] S. Payrovi, S. Babaei and E.S. Sevim, On the compressed essential graph of a commutative ring, Bull. Belg. Math. Soc. Simon Stevin 26(3), 421-429, 2019.
• [39] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31(9), 4425-4443, 2003.
• [40] K.H. Rosen, Discrete Mathematics and Its Applications, 7th ed. McGraw-Hill, 2011.
• [41] R.Y. Sharp, Steps in commutative algebra (No. 51), Cambridge university press, 2000.
• [42] J. Von Neumann, On regular rings, Proc. Natl. Acad. Sci. USA 22(12), 707-713, 1936.
• [43] E. Yaraneri, Intersection graph of a module, J. Algebra Appl. 12(05), 1250218, 2013.