The Sum Graph of Non-essential Submodules
Subject Areas : Statistics
S. Rajaee
1
(Department of Mathematics, Faculty of Basic Science, Payame Noor University, Tehran, Iran)
Keywords: گراف کامل, زیرمدول اساسی, زیرمدول کوچک, مدول همضرب,
Abstract :
Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natural numbers n, where Ω(M) is a connected graph, complete graph and has a cyclic. We prove that for a square-free natural number n, Ω(Z_n) is a complete graph. In particular, if n be the product of s distinct prime numbers, then Ω(Z_n) is the complete graph K_s. In addition, we introduce the extended graph Ω_T (M) of Ω(M) for some proper submodule T of M and we investigate about it. Dullay, we define the graph Λ(M) of module M with the set of vertices contain all nontrivial non-small submodules of M. Two distinct vertices N and K are adjacent in Λ(M) if and only if N∩K is a proper non-small submodule of M or N∩K=∘. We prove that, if M be a strongly comultiplication module, then there exists an isomorphism graph Ω(R)≅Λ(M) .
[1] Y. AL.Shaniafi, P.F. Smith, Comultiplication modules over commutative rings, Journal of commutative algebra, 3 (1) (2011), 1-29.
[2] T. Albu, G.F. Birkenmeier, A. Erdogan, A. Tercan, Ring and Module Theorey. (2009).
[3] D.D. Anderson, Some remarks on multiplication ideals II. Comm. in Algebra 28 (2000), 2577-2583.
[4] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules. Graduate Texts in Mathematics, Vol. 13, 2nd ed., Springer-Verlag, (1992).
[5] H. Ansari-Toroghy, F. Farshadifar, The dual notion of multiplication modules, Taiwanese J. Math. 11 (4) (2007), 1189-1201.
[6] H. Ansari-Toroghy, F. Farshadifar, On comultiplication modules. Korean Ann Math, 2008, 25(2): 57–66
[7] H. Ansari-Toroghy, F. Farshadifar, F. Mahboobi-Abkenar, The small intersection graph relative to multiplication modules. Journal of Algebra and Related Topics 4 (21) (2016), 21-32.
[8] H. Ansari-Toroghy, F. Mahboobi-Abkenar, The large sum graph relative to comultiplication modules, arxiv: 1609.00955 v1, 2016.
[9] A. Barnard, Multiplication modules,
J.Algebra 71 (1981), 74-178.
[10] J. Beachy, Some aspects of noncommutative localization, in Non-commutative Ring Theory, Kent State University, Lecture Notes in Mathematics, Vol. 545, Springer-Verlag, Berlin-New York, 1975.
[11] M. Baser, N. Agayev, On reduced and semicommutative modules, Turkish J. Math. 30 (2006), 285-291.
[12] Z.A. El-Bast, P.F. Smith, Multiplication modules, Comm. Algebra 16 (1988) 755-779.
[13] N.V. Dung, D.V. Huynh, P.F. Smith, R. Wisbauer, Extending modules, Pitman Research Note in Math, Series 313 Longman Harlow, 1994.
[14] F. Kasch, Modules and Rings, Academic Press, London, New York, 1982.
[15] Golan JS: Quasi-Semiperfect modules. Quart J.Math., Oxford 1971, 173-182.
[16] KR. Goodearl, Ring theory, Non-singular rings and modules. New York and Basel: Marcel Dekker INC, 1976.
[17] A. Nikseresht, H. Sharif, On comultiplication and R-multiplication modules, Journal of Algebraic systems, 2 (1) (2014), 1-19.
[18] S. Rajaee, Quasi-copure submodules, Canad. Math. Bull. 59 (2016), 197-203.
[19] S. Safaeeyan, N.Saboori Shirazi, Essential Submodules with respect to an Arbitrary Submodule, Journal of Mathematical Extension, Vol. 7, No. 3,
(2013), 15-27.
[20] S. Safaeeyan, Strongly duo and comultiplication modules, Journal of Algebraic Systems, Vol. 3, No. 1, (2016), 53-64.
[21] P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995), 124-127.
[22] P.F.Smith, M.R.Vedadi, Modules with chain conditions on Non-essential submodules, Communication in Algebra, 32 (5) (2004), 1881-1894.
[23] Y. Wang, Y. Liu, A Note on Comultiplication Modules, Algebra Colloquium 21 (1) (2014) 147-150.
