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 Γ_{reg}(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(Γ_{reg}(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.
[1] Akbari, S. Kiani, D. Mohammadi, F. Moradi, S. 2009 The total graph and regular graph of a commutative ring J. Pure Appl. Alg. 213 2224–2228 .
[2] Anderson, D. F. Livingston, P. S. 1999 The zero-divisor graph of a commutative ring J. Algebra 217 434–447 .
[3] Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley Publishing Company (1969).
[4] Azarpanah, F. Motamedi, M. 2005 , Zero-divisor graph of C[X] Acta Math. Hungar. 108 25–36 . (Ahvaz).
[5] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge University Press (1997).
[6] Chakrabarty, I. Ghosh, S. Mukherjee, T. K. Sen, M. K. 2009 Intersection graphs of ideals of rings Discrete. Math. 309 5381–5392 .
[7] Chen, J. Ding, N. Yousif, M. F. 2004 On Noetherian rings with essential socle J. Aust. Math. Soc. 76 39–49 .
[8] DeMeyer, F. McKenzie, T. Schneider, K. 2002 The zero-divisor graph of a commutative semigroup Semigroup Forum 65 206–214 .
[9] Lam, T. Y. 1991 A First Course in Non-Commutative Rings Springer-Verlag New York, Inc.
[10] Sharma, P. D. Bhatwadekar, S. M. 1995 A note on graphical representation of rings J. Algebra 176 124–127 .
[11] West, D. B. 2001 Introduction to Graph Theory 2 Prentice Hall Upper Saddle River.