DOMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH OF MODULES OVER COMMUTATIVE RINGS

Year 2021, Volume: 30 Issue: 30, 203 - 216, 17.07.2021

Habibollah Ansarı-toroghy Shokoufeh Habıbı

https://doi.org/10.24330/ieja.969902

Cited By: 1

Abstract

Let $M$ be a module over a commutative ring $R$. The
annihilating-submodule graph of $M$, denoted by $AG(M)$, is a
simple undirected graph in which a non-zero submodule $N$ of $M$
is a vertex if and only if there exists a non-zero proper
submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product
of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct
vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This
graph is a submodule version of the annihilating-ideal graph and
under some conditions, is isomorphic with an induced subgraph of
the Zariski topology-graph $G(\tau_T)$ which was introduced in [H.
Ansari-Toroghy and S. Habibi, Comm. Algebra, 42(2014), 3283-3296].
In this paper, we study the domination number of $AG(M)$ and some
connections between the graph-theoretic properties of $AG(M)$ and
algebraic properties of module $M$.

Keywords

Commutative ring, annihilating-submodule graph, domination number

References

• G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math., 312(17) (2012), 2620-2626.
• G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math., 43(5) (2013), 1415-1425.
• G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloq., 21(2) (2014), 249-256.
• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974.
• D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
• H. Ansari-Toroghy and F. Farshadifar, Product and dual product of submodules, Far East J. Math. Sci., 25(3) (2007), 447-455.
• H. Ansari-Toroghy and S. Habibi, The Zariski topology-graph of modules over commutative rings, Comm. Algebra, 42(8) (2014), 3283-3296.
• H. Ansari-Toroghy and S. Habibi, The annihilating-submodule graph of modules over commutative rings II, Arab. J. Math., 5(4) (2016), 187-194.
• H. Ansari-Toroghy and S. Habibi, The annihilating-submodule graph of modules over commutative rings, Math. Rep. (Bucur.), 20(70)(3) (2018), 245-262.
• H. Ansari-Toroghy and S. Habibi, The Zariski topology-graph of modules over commutative rings II, Arab. J. Math., 10(1) (2021), 41-50.
• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
• R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer-Verlag, New York, 2000.
• I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
• M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl., 10(4) (2011), 727-739.
• R. Diestel, Graph Theory, Third edition, Graduate Texts in Mathematics, 173, Springer-Verlag, Berlin, 2005.
• T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs Advanced Topics: Volume 2: Advanced Topics (1st ed.), Routledge, 1998.
• T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.
• C.-P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Paul., 33(1) (1984), 61-69.
• D. A. Mojdeh and A. M. Rahimi, Dominating sets of some graphs associated to commutative rings, Comm. Algebra, 40(9) (2012), 3389-3396.
• R. Nikandish, H. R. Maimani and S. Kiani, Domination number in the annihilating-ideal graphs of commutative rings, Publ. Inst. Math. (Beograd) (N.S.), 97(111) (2015), 225-231.
• R. Y. Sharp, Steps in Commutative Algebra, London Mathematical Society Student Texts, 19, Cambridge University Press, Cambridge, 1990.
• T. Tamizh Chelvam and K. Selvakumar, Central sets in the annihilating-ideal graph of commutative rings, J. Combin. Math. Combin. Comput., 88 (2014) 277-288.