For a commutative ring R with identity, the ideal-based zero-divisor graph, denoted by ?I (R), is the graph whose vertices are {x ? R\I|xy ? I for some y ? R\I}, and two distinct vertices x and y are adjacent if and only if xy?I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ?Ann(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ?Ann(M) (R) and the minimal prime ideals of Ann(M). In addition, we determine when ?Ann(M)(R) is complete. In particular, we prove that for a reduced R-module M, ?Ann(M) (R) is a complete graph if and only if R ? Z2?Z2 and M ? M1?M2 for M1 and M2 nonzero Z2-modules.

Let R be a commutative ring with identity. Let M be an R-module and T (M )

Let R be a commutative ring with identity. Let M be an R-module and T(M) * be the set of nonzero torsion elements. The set T(M) * makes up the vertices of the corresponding torsion graph, Γ R (M), with two distinct vertices x, y ∈ T(M) * forming an edge if Ann(x) ∩ Ann(y) 0. In this paper we study the case where the torsion graph Γ R (M) is planar.

In this paper, we introduce the graph G(S) of a bounded semilattice S, which is a generalization of the intersection graph of the substructures of an algebraic structure. We prove some general theorems about these graphs; as an example, we show that if S is a product of three or more chains, then G(S) is Eulerian if and only if either the length of every chain is even or all the chains are of length one. We also show that if G(S) contains a cycle, then girth(G(S)) = 3. Finally, we show that if (S,+,·,0,1) is a dually atomic bounded distributive lattice whose set of dual atoms is nonempty, and the graph G(S) of S has no isolated vertex, then G(S) is connected with diam(G(S)) ≤ 4.2000 Mathematics Subject Classification. 05C99; 06A12.

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