# A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

@article{Safaeeyan2014AGO, title={A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES}, author={Saeed Safaeeyan and Mohammad Reza Baziar and Ehsan Momtahan}, journal={Journal of Korean Medical Science}, year={2014}, volume={51}, pages={87-98} }

Abstract. Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F.Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in thepresent article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3.We also show that for a reduced module M with Z(M) ∗ 6= M \ {0},gr(Γ(M)) = ∞ if and…

## 13 Citations

### THE ZERO-DIVISOR GRAPH OF A MODULE

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Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known…

### A generalization of the zero-divisor graph for modules

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Let R be a commutative ring and M a Noetherian R-module. The zero-divisor
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Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M) is a simple undirected graph whose vertex set is Z(M)\AnnR(M) and two…

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Let M be an R-module, where R is a commutative ring with identity 1 and let G(V, E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three…

### A conception of zero-divisor graph for categories of modules

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We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R-MOD. The vertices of Γ(R-MOD) consist of all nonzero morphisms in R-MOD which are not isomorphisms. Two…

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Abstract Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices…

### THE ANNIHILATOR GRAPH FOR MODULES OVER COMMUTATIVE RINGS

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Let R be a commutative ring and M be an R-module. The annihilator graph of M , denoted by AG(M) is a simple undirected graph associated to M whose the set of vertices is ZR(M) \ AnnR(M) and two…

### Zero-divisor graphs for modules over integral domains

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The zero-divisor graphs of modules introduced and studied in [S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc. 51(1) (2014)…

### The annihilator graph of modules over commutative rings

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Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, andif $min M$, then let $I_m=(Rm:_R M)={rin R : rMsubseteq Rm}$. The annihilator graph of $M$ is…

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