g-Small Intersection Graph of a Module
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Abstract
Let be a commutative ring with identity, and be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of are studied. Properties of such as connectedness, and completeness are considered. Besides, the girth and the diameter of are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph is complete if, is a generalized hollow module or is a direct sum of two simple modules, is proved.
Received 19/04/2023
Revised 30/07/2023
Accepted 01/08/2023
Published Online First 20/01/2024
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References
Chakrabarty I, Ghosh S, Mukherjee TK, Sen MK. Intersection Graphs of Ideals of Rings. Discrete Math. 2009; Vol. 309: 5381-5392. https://doi.org/10.1016/j.endm.2005.06.104
Akbari S, Tavallaee HA, Khalashi Ghezelahmad S. Intersection Graph of Submodules of a Module. J Algebra Appl. 2012; 11(1): 1250019, 1-8. https://doi.org/10.1142/S0219498811005452
Alwan AH. Maximal Ideal Graph of Commutative Semirings. Int J Nonlinear Anal Appl. 2021; 12(1): 913-926. https://doi.org/10.22075/ijnaa.2021.4946
Alwan AH. Maximal Submodule Graph of a Module. J Discrete Math Sci Cryptogr. 2021; 24(7): 1941-1949. https://doi.org/10.1080/09720529.2021.1974652
Amreen J, Naduvath S. Order Sum Graph of a Group. Baghdad Sci J. 2023; 20(1): 181-188. http://dx.doi.org/10.21123/bsj.2022.6480
Mahdavi LA, Talebi Y. On the small intersection graph of submodules of a module. Algebr Struct their Appl. 2021; 8(1): 117-130. https://civilica.com/doc/1580010
Wisbauer R. Foundations of Module and Ring Theory. Gordon and Breach, Reading; 1991. https://doi.org/10.1201/9780203755532
Koşar B, Nebiyev C, Sokmez N. G-Supplemented Modules. Ukr Math. J. 2015; 67(6): 861-864. https://doi.org/10.1007/s11253-015-1127-
Zhou DX, Zhang XR. Small-Essential Submodules and Morita Duality. Southeast Asian Bull. Math. 2011; 35(6): 1051-1062. http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201106&filename=14_35(6).pdf
Koşar B, Nebiyev C, Pekin A. A Generalization of g-Supplemented Modules. Miskolc Math Notes. 2019; 20(1): 345-352. https://doi.org/10.18514/MMN.2019.2586
Bondy JA, Murty USR. Graph Theory. Graduate Texts in Mathematics 244. New York: Springer; 2008.
Ӧkten HH, Pekin A. On g-Radical Supplement Submodules. Miskolc Math Notes. 2021; 22(2): 687-693. DOI: https://doi.org/10.18514/MMN.2021.3394
Kaynar E, Türkmen E, Çalışıcı H. SS-Supplemented Modules. Commun Fac Sci Univ Ank Sér. A1, Math. Stat. 2020; 69(1): 473-485. https://doi.org/10.31801/cfsuasmas.585727
Clark J, Lomp C, Vanaja N, Wisbauer R. Lifting Modules, Supplements and Projectivity in Module Theory. Frontiers in Mathematics, Birkauser Verlag; 2006. https://link.springer.com/book/10.1007/3-7643-7573-6
Al-Harere MN, Mitlif RJ, Sadiq FA. Variant Domination Types for a Complete h-Ary Tree. Baghdad Sci J. 2021; 18(1(Suppl.)): 2078-8665. https://doi.org/10.21123/bsj.2021.18.1(Suppl.).0797