Associated graphs of le-modules


Sadashiv Ramkrushna Puranik
Sachin Ballal
Vilas Kharat


Let M be an le-module over a commutative ring with unity. In this paper, an associated graph G(M) of M with all nonzero proper submodule elements of M as vertices has been introduced and studied. Any two distinct vertices n and m are adjacent if n +m = e. Some algebraic, topological and, graph theoretic properties of le-modules have been established. Also, it is shown that the Beck’s conjecture is true for coatomic le-modules.


How to Cite
Sadashiv Ramkrushna Puranik, Sachin Ballal, & Vilas Kharat. (2021). Associated graphs of le-modules. International Journal of Next-Generation Computing, 12(2), 280–291.


  1. A. Abbasi, H. Roshan-Shekalgourabi, D. Hassanzadeh-Lelekaami Associated Graphs of Modules Over Commutative Rings, Iranian Journal of Mathematical Sciences and Informatics, 10(1)(2015), pp 45-58.
  2. A. K. Bhuniya and M. Kumbhakar, On irrducible pseudo-prime spectrum of topologicalle-modules, Quasigroups and Related Systems ,26(2)(2018)251-262.
  3. A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laske-rian le-modules, Acta Math. Hunga. , 158(1)(2019), 202-215.
  4. A. K. Bhuniya and M. Kumbhakar, On the prime spectrum of an lemodule , Journal of Algebra and its Applications, DOI: 10.1142/S0219498821502200.
  5. A. K. Bhuniya and M. Kumbhakar, On multiplication le-modules, DOI:10.13140/RG.2.2.20913.74082.
  6. B. Csakany, G. Pollak, The graph of subgroups of a nite group, Czechoslovak Mathematical Journal, 19(94)(1969), 241-247
  7. Elham Mehdi-Nezhad and Amir M. Rahimi, Comaximal Submodule Graphs of Unitary Modules, Libertas Mathematica (new series),37(1), 75-98.
  8. George Gratzer, Lattice Theory: Foundation, Birkhauser, Springer, 2010.
  9. I. Beck, Coloring of a commutative ring, J. Algebra., 116 (1988), 208-226.
  10. Ivy Chakrabarty, ShamikGhosh, T.K.Mukherjee, M.K.Sen, Intersection graphs of ideals of rings, Discrete Mathematics, 309(17), 5381-5392.
  11. J. A. Bondy and U. S. R. Murty, Graph theory, Gradute Text in Mathematics, 244, Springer, New York, 2008.
  12. J.R. Munkres , Topology, Second Ed. , Prentice Hall, New Jersey, (1999).
  13. Narayan Phadatare, Sachin Ballal and Vilas Kharat, Semi-complement graph of lattice modules, Springer-Verlag GmbH Germany, Soft Computing (2019) 23:3973-3978. doi:org/10.1007/s00500-018-3347-y
  14. Narayan Phadatare, Sachin Ballal and Vilas Kharat, On the graph of multipli-cation lattice modules, Eurasian Bulletin of Mathematics , 1(3)(2018), 117-125.
  15. Sachin Ballal and Vilas Kharat, Zariski topology on lattice modules, Asian-European Journal of Mathematics , 8(4) (2015) 1550066 (10 pages)
  16. Sachin Ballal and Vilas Kharat,On minimal spectrum of multiplication lattice modules, Mathematica Bohemica, 144(1)(2019), 85-97.
  17. Sachin Ballal and Vilas Kharat, On generalization of prime, weakly prime and almost prime elements in multiplicative lattices, Int. J. Algebra, 8(9), 439-449.
  18. Sachin Ballal and Vilas Kharat, On-Absorbing Primary Elements in Lattice Modules, Algebra, (2015), 1-6.
  19. Sachin Ballal, M. Gophane and Vilas Kharat, On Weakly Primary Elements in Multiplicative Lattices, Southeast Asian Bulletin of Mathematics, 40(1)(2016),49-57.
  20. T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005.
  21. V. Joshi and S. Ballal, A note on n-Baer multiplicative lattices, Southeast Asian Bull. Math. 39 (2015), 67-76.