RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

Let R be a commutative ring and A(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A\{(0)} and two distinct vertices I and J are adjacent if and only if I J = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus gamma(AG(R)). It is shown that if R is an Artinian ring such that gamma(AG(R)) < infinity, then either R has only finitely many ideals or (R, m) is a Gorenstein ring with maximal ideal m and v.dim(R/m)m/m(2) = 2. Also, for any two integers g >= 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) gamma(AG(R)) = g and (ii) vertical bar R/m vertical bar <= q for every maximal ideal m of R. Also, it is shown that if R is a non-domain Noetherian local ring such that gamma(AG(R)) < infinity, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.