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The total graph of non-zero annihilating ideals of a commutative ring. (English) Zbl 1401.13023

Summary: Assume that \(R\) is a commutative ring with non-zero identity which is not an integral domain. An ideal \(I\) of \(R\) is called an annihilating ideal if there exists a non-zero element \(a\in R\) such that \(Ia=0\). S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of \(R\), denoted by \(\Omega(R)\), as the graph with the vertex-set \(\mathrm{A}(R)^\ast\), the set of all non-zero annihilating ideals of \(R\), and two distinct vertices \(I\) and \(J\) are adjacent if \(I+J\) is an annihilating ideal. In this paper, we study the relations between the diameters of \(\Omega(R)\) and \(\Omega(R[[x]])\). Also, we study the relations between the diameters of \(\Omega(R)\) and \(\Omega(R[[x]])\), whenever \(R\) is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of \(\Omega(R)\) such as domination number and independence number. Furthermore, we study the complement of this graph.

MSC:

13A99 General commutative ring theory
05C75 Structural characterization of families of graphs
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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