Year 2021,
, 1658 - 1666, 14.12.2021
Ece Yetkin Çelikel
,
Angsuman Das
,
Cihat Abdioğlu
References
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rings, Commun. Algebra 43 (6), 2457-2465, 2015.
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Algebra 217, 434-447, 1999.
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graphs of commutative rings, Asian-Eur. J. Math. 12 (6), 1-19, 2019. (Article ID
2040003)
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New York, 1998.
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- [7] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math.,
1992.
- [8] G. Hahn and C. Tardif, Graph Homomorphisms: Structure and Symmetry, in: Graph
Symmetry, 107-166, Springer, Dordrecht, 1997.
- [9] I. Kaplansky, Commutative Rings (rev. ed.), University of Chicago Press, Chicago,
1974.
- [10] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com-
mutative rings, Commun. Algebra 40 (9), 3389-3396, 2012.
- [11] N.J. Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs, Canad.
Math. Bull. 56 (2), 407-411, 2013.
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Algebra 31 (9), 4425-4443, 2003.
Ideal-based quasi zero divisor graph
Year 2021,
, 1658 - 1666, 14.12.2021
Ece Yetkin Çelikel
,
Angsuman Das
,
Cihat Abdioğlu
Abstract
Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.
References
- [1] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, The inclusion ideal graph of
rings, Commun. Algebra 43 (6), 2457-2465, 2015.
- [2] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J.
Algebra 217, 434-447, 1999.
- [3] S. Aykaç, N. Akgüne and A.S. Çevik, Analysis of Zagreb indices over zero-divisor
graphs of commutative rings, Asian-Eur. J. Math. 12 (6), 1-19, 2019. (Article ID
2040003)
- [4] I. Beck, Coloring of commutative rings, J. Algebra 116, 208-226, 1988.
- [5] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, Springer-Verlag,
New York, 1998.
- [6] L. Fuchs, On quasi-primary ideals, Acta. Sci. Math. (Szeged) 11 (3), 174-183, 1947.
- [7] R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Appl. Math.,
1992.
- [8] G. Hahn and C. Tardif, Graph Homomorphisms: Structure and Symmetry, in: Graph
Symmetry, 107-166, Springer, Dordrecht, 1997.
- [9] I. Kaplansky, Commutative Rings (rev. ed.), University of Chicago Press, Chicago,
1974.
- [10] D.A. Mojdeh and A.M. Rahimi, Dominating sets of some graphs associated to com-
mutative rings, Commun. Algebra 40 (9), 3389-3396, 2012.
- [11] N.J. Rad, S.H. Jafari and D.A. Mojdeh, On domination in zero-divisor graphs, Canad.
Math. Bull. 56 (2), 407-411, 2013.
- [12] S.P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun.
Algebra 31 (9), 4425-4443, 2003.