Acta Mathematica Hungarica
Volume/Issue:
Volume 134: Issue 4
The regular digraph of ideals of a commutative ring
Authors: M. J. Nikmehr 1 and F. Shaveisi 1
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  • 1 Department of Mathematics, Faculty of Science, K.N. Toosi University of Technology, Tehran, Iran
DOI:
https://doi.org/10.1007/s10474-011-0139-6
Page Count:
516–528
Publication Date:
01 Mar 2012
Online Publication Date:
22 Jul 2011
Article Category:
Research Article
Restricted access

Abstract

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≦ω(Γreg(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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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
regular digraph; clique number; chromatic number; Artinian ring; 05C20; 05C69; 13E05; 16P20

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