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THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING

Year 2020, , 61 - 76, 07.01.2020
https://doi.org/10.24330/ieja.662957

Abstract

Let $R$ be an associative ring with $1\neq 0$ which is not a domain. Let $A(R)^*=\{I\subseteq R~|~I \text{ is a left or right ideal of } R \text{ and } \mathrm{l.ann}(I)\cup \mathrm{r.ann}(I)\neq0\}\setminus\{0\}$. The total graph of annihilating one-sided ideals of $R$, denoted by $\Omega(R)$, is a graph with the vertex set $A(R)^*$ and two distinct vertices $I$ and $J$ are adjacent if $\mathrm{l.ann}(I+J)\cup \mathrm{r.ann}(I+J)\neq0$. In this paper, we study the relations between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose graphs are disconnected. Also, we study diameter, girth, independence number, domination number and planarity of this graph.

References

  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. H. Poincare Sect. B (N.S.), 3 (1967), 433-438.
  • I. Gitler, E. Reyes and R. H. Villarreal, Ring graphs and complete intersection toric ideals, Discrete Math., 310(3) (2010), 430-441.
  • K. R. Goodearl and R. B. War eld, Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society, Student Texts, 16, Cam- bridge University Press, Cambridge, 1989.
  • C. Y. Hong, N. K. Kim and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra, 151(3) (2000), 215-226.
  • I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago, 1974.
  • N. K. Kim and Y. Lee, Extension of reversible rings, J. Pure Appl. Algebra, 185(1-3) (2003), 207-223.
  • J. Krempa, Some examples of reduced rings, Algebra Colloq., 3(4) (1996), 289- 300.
  • T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, Berlin/Heidelberg, New York, 1991.
  • S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl., 6(4) (2014), 1450047 (22 pp).
  • D. B.West, Introduction to Graph Theory, Second Edition, Prentice-Hall, Inc., Upper Saddle River, 2001.
Year 2020, , 61 - 76, 07.01.2020
https://doi.org/10.24330/ieja.662957

Abstract

References

  • D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217(2) (1999), 434-447.
  • I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208-226.
  • G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. H. Poincare Sect. B (N.S.), 3 (1967), 433-438.
  • I. Gitler, E. Reyes and R. H. Villarreal, Ring graphs and complete intersection toric ideals, Discrete Math., 310(3) (2010), 430-441.
  • K. R. Goodearl and R. B. War eld, Jr., An Introduction to Noncommutative Noetherian Rings, London Mathematical Society, Student Texts, 16, Cam- bridge University Press, Cambridge, 1989.
  • C. Y. Hong, N. K. Kim and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra, 151(3) (2000), 215-226.
  • I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago, 1974.
  • N. K. Kim and Y. Lee, Extension of reversible rings, J. Pure Appl. Algebra, 185(1-3) (2003), 207-223.
  • J. Krempa, Some examples of reduced rings, Algebra Colloq., 3(4) (1996), 289- 300.
  • T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, Berlin/Heidelberg, New York, 1991.
  • S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl., 6(4) (2014), 1450047 (22 pp).
  • D. B.West, Introduction to Graph Theory, Second Edition, Prentice-Hall, Inc., Upper Saddle River, 2001.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Abolfazl Alibemani This is me

Ebrahim Hashemi This is me

Abdollah Alhevaz This is me

Publication Date January 7, 2020
Published in Issue Year 2020

Cite

APA Alibemani, A., Hashemi, E., & Alhevaz, A. (2020). THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING. International Electronic Journal of Algebra, 27(27), 61-76. https://doi.org/10.24330/ieja.662957
AMA Alibemani A, Hashemi E, Alhevaz A. THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING. IEJA. January 2020;27(27):61-76. doi:10.24330/ieja.662957
Chicago Alibemani, Abolfazl, Ebrahim Hashemi, and Abdollah Alhevaz. “THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 61-76. https://doi.org/10.24330/ieja.662957.
EndNote Alibemani A, Hashemi E, Alhevaz A (January 1, 2020) THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING. International Electronic Journal of Algebra 27 27 61–76.
IEEE A. Alibemani, E. Hashemi, and A. Alhevaz, “THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING”, IEJA, vol. 27, no. 27, pp. 61–76, 2020, doi: 10.24330/ieja.662957.
ISNAD Alibemani, Abolfazl et al. “THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING”. International Electronic Journal of Algebra 27/27 (January 2020), 61-76. https://doi.org/10.24330/ieja.662957.
JAMA Alibemani A, Hashemi E, Alhevaz A. THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING. IEJA. 2020;27:61–76.
MLA Alibemani, Abolfazl et al. “THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 61-76, doi:10.24330/ieja.662957.
Vancouver Alibemani A, Hashemi E, Alhevaz A. THE TOTAL GRAPH OF ANNIHILATING ONE-SIDED IDEALS OF A RING. IEJA. 2020;27(27):61-76.