Research Article
Year 2020,
, 61 - 76, 07.01.2020
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
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- N. K. Kim and Y. Lee, Extension of reversible rings, J. Pure Appl. Algebra, 185(1-3) (2003), 207-223.
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- D. B.West, Introduction to Graph Theory, Second Edition, Prentice-Hall, Inc., Upper Saddle River, 2001.
Year 2020,
, 61 - 76, 07.01.2020
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 | |
| Publication Date | January 7, 2020 |
| Published in Issue | Year 2020 |