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…
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References
SHOWING 1-10 OF 20 REFERENCES
A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL
- Mathematics
- 2013
Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈…
Zero divisor graphs for modules over commutative rings
- Mathematics
- 2012
. In this article, we give several generalizations of the concept of zero-divisor elements in a commutative ring with identity to modules. Then, for each R -module M , we associate three undirected…
Zero-divisor Graph of Non-commutative Ring
- Mathematics
- 2012
The concept of zero-divisor graph Γ(R) of a commutative ring R was introduced by Beck [19]. However, he let all the elements of a commutative ring be vertices of the graph and was mainly interested…
The Classification of the Annihilating-Ideal Graphs of Commutative Rings
- Mathematics
- 2014
Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and…
The Annihilating-Ideal Graph of Commutative Rings II
- Mathematics
- 2008
In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]). Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with…