The strongly annihilating-submodule graph of a module

Document Type : Research Paper


1 Department of Mathematics, Lorestan University, Khorramabad, Iran

2 Department of Mathematics, Lorestan university, Khorramabad, Iran


In this paper, we define the notion of strongly annihilating-submodule graph of modules. This graph is a straightforward common generalization of the annihilating-submodule graph  and the annihilating-ideal graph. In addition to providing the properties of this graph in general, we investigate the behavior of the graph when modules are reduced or divisible.


[1] S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra, 270(1) (2003), 169-180.
[2] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra, 296(2) (2006), 462-479.
[3] D. F. Anderson, R. Levy and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221-241.
[4] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
[5] D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra Appl., 159(2) (1993), 500-514.
[6] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, New York: Springer-Verlag, (1992).
[7] H. Ansari-Toroghy and Sh. Habibi, The annihilating-submodule graph of modules over commutative rings
II, Arab. j. Math., 5(4) (2016), 187-194.
[8] H. Ansari-Toroghy and Sh. Habibi, The Zariski topology-graph of modules over commutative rings, Comm.
Algebra, 42(8) (2014), 3283-3296.
[9] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl., 10(4)
(2011), 727-739.
[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl., 10(4)
(2011), 741-753.
[11] S. Ceken, M. Alkan and P. F. Smith, Second modules over noncommutative rings, Comm. Algebra, 41(1)
(2013), 83-98.
[12] R. Diestel, Graph Theory, Electronic Edition, New York: Springer-Verlag, Heidelberg, (1997), (2000),
[13] Z. A. EI-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755-779.
[14] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Math., New York, Heid-elberg-Berlin:
Springer-Verlag, (1999).
[15] S. Safaeeyan, E. Momtahan and M. Baziar, Zero-divisor graphs for modules over integral domains, J.
Algebra Appl., 16(5) (2017), 8 pages.
[16] L. Toth, Subgroups of nite abelian groups having rank via Goursat's lemma, Tatra Mt. Math. Publ., 59(1)
(2014), 93-103.