In this paper the zero-divisor graph Γ (R) of a commutative reduced ring R is studied. We associate the ring properties of R, the graph properties of Γ (R) and the topological properties of Spec(R). Cycles in Γ (R) are investigated and an algebraic and a topological characterization is given for the graph Γ (R) to be triangulated or hypertriangulated. We show that the clique number of Γ (R), the cellularity of Spec(R) and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and 2 ∈ Z(R); Γ (R) is complemented if and only if Min(R) is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ (R) is between the density and the weight of Spec(R). We show that Γ (R) is not triangulated and the set of centers of Γ (R) is a dominating set if and only if the set of isolated points of Spec(R) is dense in Spec(R).

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Abstract. Let R be a commutative ring with 1. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined a graph on R, Γ(R), with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper, we consider a subgraph Γ 2 (R) of Γ(R) which consists of non-unit elements. We investigate the behavior of Γ 2 (R) and Γ 2 (R) \ J(R), where J(R) is the Jacobson radical of R. We associate the ring properties of R, the graph properties of Γ 2 (R) and the topological properties of Max(R). Diameter, girth, cycles and dominating sets are investigated and the algebraic and the topological characterizations are given for graphical properties of these graphs.

An R-module M is called a multiplication module if for each submodule N of M, N = I M for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M). Also a finitely generated module M is a Baer module if and only if Spec(M) is an extremally disconnected space; if and only if it is a C S-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N , Ann(x) ⊆ (N : M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (V N M); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a V N M.