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.
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. Wareld, 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,
Volume: 27 Issue: 27, 61 - 76, 07.01.2020
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. Wareld, 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.
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.