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.
Highlights
In recent years, using graph theoretical tools in the study of algebraic structures attracted many researchers, see, for instance, [1,2,11]
Beck in [2] introduced the idea of a zero-divisor graph of a commutative ring, where he was mainly interested in colorings
Authors in [1] introduced the zero-divisor graph of a commutative ring R, denoted by Γ(R), as the graph with vertices Z(R)∗, the set of all nonzero zero-divisors of R, and two distinct vertices x and y are adjacent if xy = 0
Summary
In recent years, using graph theoretical tools in the study of algebraic structures attracted many researchers, see, for instance, [1,2,11]. Since Ra is adjacent to I1 and R is a prime ring, I1 is not a right ideal. Since R is a nonprime ring, there exists a nonzero two-sided ideal K such that l.ann(K) = 0.
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