On A Graph Of Submodules

Year 2019, Volume: 23 Issue: 3, 396 - 402, 01.06.2019

Ali Öztürk , Tahire Özen , Erol Yılmaz

https://doi.org/10.16984/saufenbilder.483138

Abstract

Let
S be an assosiative ring with identitiy and N be a right S-module. We define
the non-maximal graph m(N) of N with all non-trivial submodules of N as vertices
and two distinct vertices  A,B are
adjecent if and only if A + B is not maximal submodule of N. In this paper, we
investigate the connectivity, completeness,  girth, domination nuber, cut edges,
perfectness and r-partite of m(N). Moreover,  we
give connections between the graph-theoretic properties of m(N) and algebraic properties of
N.

Keywords

Non-maximal submodule, connected and copmlete graph

References

• [1] S. Akbari, H.A. Tavallae and S. Khalashi Gheze-lahmad, “Intersection graphs of submodules of a modules,” J. Algebra and Its Aplications, vol. 11, no. 1, 2012, 1250019.[2] F.W. Anderson and K.R. Fuller, Rings and Cat-egories of Modules, Springer, New York, 1992.[3] J.A. Bandy and U.S.R. Murty, Graph Theory, Springer, New York, 2008.[4] I. Beck, “Coloring of a commutative ring,” J. Algebra, vol. 116, pp. 208-226, 1998.[5] I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen, “Intersection graphs of ideals of rings,” Discrete Math., vol. 309, pp. 5381-5392, 2009. [6] Loft Ali Mahdavi and Yahya Talebi, “Co-intersection graph of submodules of a mod-ule,” Algebra and Discrete Mathematics, vol. 1, pp. 128-143, 2016.[7] E. Yaraneri, “Intersection graphs of modules,” J. Algebra Appl., vol. 12, 2013, 1250218.