ON THE STRONGLY ANNIHILATING-SUBMODULE GRAPH OF A MODULE
Year 2022,
, 443 - 454, 01.04.2022
Reza Beyranvand
,
Ahadollah Farzı-safarabadı
References
- [1] 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 Col-
loq. 21 (2), 249-256, 2014.
- [2] 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, 2620-2626,
2012.
- [3] S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a
complete r-partite graph, J. Algebra, 270, 169-180, 2003.
- [4] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J.
Algebra, 296, 462-479, 2006.
- [5] F. Aliniaeifard, M. Behboodi, E. Mehdi-nezhad and A.M. Rahimi, The annihilating-
ideal graph of a commutative ring with respect to an ideal, Comm. Algebra , 42,
2269-2284, 2014.
- [6] D.F. Anderson, R. Levy and J. Shapiro J, Zero-divisor graphs, von Neumann regular
rings, and Boolean algebras, J. Pure Appl. Algebra, 180, 221-241, 2003.
- [7] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J.
Algebra , 217, 434-447, 1999.
- [8] D.D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra
Appl. 159, 500-514, 1993.
- [9] H. Ansari-Toroghy and Sh. Habibi, The Zariski topology-graph of modules over com-
mutative rings, Comm. Algebra 42, 3283-3296, 2014.
- [10] H. Ansari-Toroghy and Sh. Habibi, The annihilating-submodule graph of modules over
commutative rings II. Arab J. Math. 5, 187-194, 2016.
- [11] H. Ansari-Toroghy and Sh. Habibi, The annihilating-submodule graph of modules over
commutative rings, Math. Reports 20, 245-262, 2018.
- [12] A. Azizi, Weakly prime submodule and prime submodule, Glasgow Math. J. 48, 343-
346, 2006.
- [13] I. Beck, Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
- [14] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J.
Algebra Appl. 10 (4), 727-739, 2011.
- [15] R. Beyranvand and A. Farzi-Safarabadi, The strongly annihilating-submodule graph
of a module, Algebraic Struc. Appl. 7 (1), 83-99, 2020.
- [16] R. Beyranvand and F. Rastgoo, Weakly second modules over noncommutative rings,
Hacet. J. Math. Stat. 45 (5), 1355-1366, 2016.
- [17] S. Ceken, M. Alkan and P.F. Smith, Second modules over noncommutative rings,
Comm. Algebra, 41, 83-98, 2013.
- [18] F.R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commu-
tative semigroup, Semigroup Forum, 65, 206-214, 2002.
- [19] R. Diestel, Graph Theory. Electronic Edition, New York: Springer-Verlag Heidelberg
1997, 2000, 2005.
- [20] N. Jafari Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs,
Canad. Math. Bull. 56 (2), 407-411, 2013.
- [21] T.Y. Lam Lectures on modules and rings, Graduate Texts in Math. New York
Heidelberg-Berlin: Springer-Verlag 1999.
- [22] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com-
mutative rings, Comm. Algebra, 40, 3389-3396, 2012.
- [23] R. Nikandish, H. Maimani and S. Kiani, Domination numer in the annihilating-ideal
graphs of commutative rings, Publications De L’institut Math. 97 (111), 225-231,
2015.
- [24] S. Safaeeyan, Annihilating submodule graph for modules, Trans. Comb. 7 (1), 1-12,
2018.
- [25] S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph
for modules, J. Korean Math. Soc. 51 (1), 87-98, 2014.
- [26] L. Toth, Subgroups of finite abelian groups having rank two via Goursats lemma, Tatra
Mt. Math. Publ. 59 (1), 93-103, 2014.
On the strongly annihilating-submodule graph of a module
Year 2022,
, 443 - 454, 01.04.2022
Reza Beyranvand
,
Ahadollah Farzı-safarabadı
Abstract
In this paper we continue to study the strongly annihilating-submodule graph. In addition to providing the more properties of this graph, we compare extensively the properties of this graph with the annihilating-submodule graph.
References
- [1] 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 Col-
loq. 21 (2), 249-256, 2014.
- [2] 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, 2620-2626,
2012.
- [3] S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a
complete r-partite graph, J. Algebra, 270, 169-180, 2003.
- [4] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J.
Algebra, 296, 462-479, 2006.
- [5] F. Aliniaeifard, M. Behboodi, E. Mehdi-nezhad and A.M. Rahimi, The annihilating-
ideal graph of a commutative ring with respect to an ideal, Comm. Algebra , 42,
2269-2284, 2014.
- [6] D.F. Anderson, R. Levy and J. Shapiro J, Zero-divisor graphs, von Neumann regular
rings, and Boolean algebras, J. Pure Appl. Algebra, 180, 221-241, 2003.
- [7] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J.
Algebra , 217, 434-447, 1999.
- [8] D.D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra
Appl. 159, 500-514, 1993.
- [9] H. Ansari-Toroghy and Sh. Habibi, The Zariski topology-graph of modules over com-
mutative rings, Comm. Algebra 42, 3283-3296, 2014.
- [10] H. Ansari-Toroghy and Sh. Habibi, The annihilating-submodule graph of modules over
commutative rings II. Arab J. Math. 5, 187-194, 2016.
- [11] H. Ansari-Toroghy and Sh. Habibi, The annihilating-submodule graph of modules over
commutative rings, Math. Reports 20, 245-262, 2018.
- [12] A. Azizi, Weakly prime submodule and prime submodule, Glasgow Math. J. 48, 343-
346, 2006.
- [13] I. Beck, Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
- [14] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J.
Algebra Appl. 10 (4), 727-739, 2011.
- [15] R. Beyranvand and A. Farzi-Safarabadi, The strongly annihilating-submodule graph
of a module, Algebraic Struc. Appl. 7 (1), 83-99, 2020.
- [16] R. Beyranvand and F. Rastgoo, Weakly second modules over noncommutative rings,
Hacet. J. Math. Stat. 45 (5), 1355-1366, 2016.
- [17] S. Ceken, M. Alkan and P.F. Smith, Second modules over noncommutative rings,
Comm. Algebra, 41, 83-98, 2013.
- [18] F.R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commu-
tative semigroup, Semigroup Forum, 65, 206-214, 2002.
- [19] R. Diestel, Graph Theory. Electronic Edition, New York: Springer-Verlag Heidelberg
1997, 2000, 2005.
- [20] N. Jafari Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs,
Canad. Math. Bull. 56 (2), 407-411, 2013.
- [21] T.Y. Lam Lectures on modules and rings, Graduate Texts in Math. New York
Heidelberg-Berlin: Springer-Verlag 1999.
- [22] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com-
mutative rings, Comm. Algebra, 40, 3389-3396, 2012.
- [23] R. Nikandish, H. Maimani and S. Kiani, Domination numer in the annihilating-ideal
graphs of commutative rings, Publications De L’institut Math. 97 (111), 225-231,
2015.
- [24] S. Safaeeyan, Annihilating submodule graph for modules, Trans. Comb. 7 (1), 1-12,
2018.
- [25] S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph
for modules, J. Korean Math. Soc. 51 (1), 87-98, 2014.
- [26] L. Toth, Subgroups of finite abelian groups having rank two via Goursats lemma, Tatra
Mt. Math. Publ. 59 (1), 93-103, 2014.