2014/11/15 by F. Aliniaeifard, M. Behboodi, Aliniaeifard, F. +5
Mathematics · #05C38 #13B25 #13F20 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:05C38 #msc:13B25 #msc:13F20
paper · pdf · doi:10.48550/arxiv.1411.4163
11 pages, 1 figure
arxiv created 2014/11/15 · arxiv updated 2014/11/18
Let R be a commutative ring with 1≠ 0 and \BbbA(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph \BbbAG(R) with the vertex set \BbbA(R)* = \BbbA(R)∖ \(0)\ and two distinct vertices I and J are adjacent if and only if IJ = (0). In this paper, we first study the interplay between the diameter of annihilating-ideal graphs and zero-divisor graphs. Also, we characterize rings R when \rm gr(\BbbAG(R))≥ 4, and so we characterize rings whose annihilating-ideal graphs are bipartite. Finally, in the last section we discuss on a relation between the Smarandache vertices and diameter of \Bbb AG(R).