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  • 1 Department of Mathematics, Faculty of Science, K.N. Toosi University of Technology, Tehran, Iran
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Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by , is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max (R)|−1≦ωreg(R))≦|Max (R)| and , where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.

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