2015/01/18 by Farid Aliniaeifard, Mahmood Behboodi, Aliniaeifard, Farid +3
Mathematics · #05C10 #13E10 #13H99 #16P60 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:05C10 #msc:13E10 #msc:13H99 #msc:16P60
paper · pdf · doi:10.48550/arxiv.1501.04329
9 pages, 3 figures. arXiv admin note: text overlap with arXiv:1102.4835
arxiv created 2015/01/18 · arxiv updated 2015/01/20
Let R be a commutative ring and \BbbA(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph \BbbAG(R) with vertex set \BbbA(R)^*=\BbbA∖\(0)\ such that two distinct vertices I and J are adjacent if and only if IJ=(0). We characterize commutative Noetherian rings R whose annihilating-ideal graphs have finite genus γ(\BbbAG(R)). It is shown that if R is a Noetherian ring such that 0<γ(\BbbAG(R))<∞, then R has only finitely many ideals.