We survey the research conducted on zero divisor graphs, with a focus on zero divisor graphs determined by equivalence classes of zero divisors of a commutative ring R. In particular, we consider the problem of classifying star graphs with any finite number of vertices. We study the pathology of a zero divisor graph in terms of cliques, we investigate when the clique and chromatic numbers are equal, and we show that the girth of a Noetherian ring, if finite, is 3. We also introduce a graph for modules that is useful for studying zero divisor graphs of trivial extensions.

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On Zero Divisor Graphs

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GTH(R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GTH(R).

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The generalized total graph of a commutative ring

In this study, we introduce the concepts of S -prime submodules and S -torsion-free modules, which are generalizations of prime submodules and torsion-free modules. Suppose S ⊆ R is a multiplicatively closed subset of a commutative ring R , and let M be a unital R -module. A submodule P of M with (P :R M) ∩ S = ∅ is called an S -prime submodule if there is an s ∈ S such that am ∈ P implies sa ∈ (P :R M) or sm ∈ P. Also, an R -module M is called S -torsion-free if ann(M) ∩ S = ∅ and there exists s ∈ S such that am = 0 implies sa = 0 or sm = 0 for each a ∈ R and m ∈ M. In addition to giving many properties of S -prime submodules, we characterize certain prime submodules in terms of S -prime submodules. Furthermore, using these concepts, we characterize some classical modules such as simple modules, S -Noetherian modules, and torsion-free modules.

On S-prime submodules

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

Idempotent and Nilpotent Submodules of Multiplication Modules

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, …- regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph i(R) of R induced by the above elements. 2000 Mathematics Subject Classiflcation: primary 13F99; secondary 16E50, 13A15

Von Neumann Regular and Related Elements in Commutative Rings

https://projecteuclid.org/download/pdf_1/euclid.jca/1229376151

M. Winders

Papers

Idealization of a Module