2011/02/23 by Farid Aliniaeifard, Mahmood Behboodi, Aliniaeifard, Farid +1 · 1 citation
Mathematics · #05C10 #13E05 #13E10 #13M05 #16P60 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.RA #msc:05C10 #msc:13E05 #msc:13E10 #msc:13M05 #msc:16P60
paper · pdf · doi:10.48550/arxiv.1102.4835
13 pages
arxiv created 2011/02/23 · arxiv updated 2011/02/24
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 the vertex set \BbbA(R)^*=\BbbA∖\(0)\ and two distinct vertices I and J are adjacent if and only if IJ=(0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(\BbbAG(R)). It is shown that if R is an Artinian ring such that γ(\BbbAG(R))<∞, then R has finitely many ideals or (R,\mathfrakm) is a Gorenstein ring with maximal ideal \mathfrakm and \rm v.dim_R/\mathfrakm\mathfrakm/\mathfrakm2=2. Also, for any two integers g≥ 0 and q>0, there are finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(\BbbAG(R)) < g and (ii) |R/\mathfrakm| ≤ q for every maximal ideal \mathfrakm of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(\BbbAG(R))<∞, then either R is a Gorenstein ring or R is an Artinian ring with finitely many ideals.