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

Authors : Abolfazl ALİBEMANİ, Ebrahim HASHEMİ, Abdollah ALHEVAZ

Pages : 61-76

Doi:10.24330/ieja.662957

View : 49 | Download : 11

Publication Date : 2020-01-07

Article Type : Research Paper

Abstract :Let $R$ be an associative ring with $1\neq 0$ which is not a domain. Let $Ainsert ignore into journalissuearticles values(R);^*=\{I\subseteq R~|~I \text{ is a left or right ideal of } R \text{ and } \mathrm{l.ann}insert ignore into journalissuearticles values(I);\cup \mathrm{r.ann}insert ignore into journalissuearticles values(I);\neq0\}\setminus\{0\}$. The total graph of annihilating one-sided ideals of $R$, denoted by $\Omegainsert ignore into journalissuearticles values(R);$, is a graph with the vertex set $Ainsert ignore into journalissuearticles values(R);^*$ and two distinct vertices $I$ and $J$ are adjacent if $\mathrm{l.ann}insert ignore into journalissuearticles values(I+J);\cup \mathrm{r.ann}insert ignore into journalissuearticles values(I+J);\neq0$. 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.
Keywords : Total graph, diameter, reversible ring, semicommutative ring, skew polynomial ring

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