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

Year 2020, , 61 - 76, 07.01.2020

Abolfazl Alibemani Ebrahim Hashemi Abdollah Alhevaz

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.

Keywords

Total graph, diameter, reversible ring, semicommutative ring, skew polynomial ring

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