Minimal prime ideals and cycles in annihilating-ideal graphs

@article{Aalipour2013MinimalPI,
  title={Minimal prime ideals and cycles in annihilating-ideal graphs},
  author={Ghodratollah Aalipour and Saieed Akbari and Reza Nikandish and Mohammad Javad Nikmehr and Farzad Shaveisi},
  journal={Rocky Mountain Journal of Mathematics},
  year={2013},
  volume={43},
  pages={1415-1425}
}
Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)∗ = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study some connections between the graph theoretic properties of this graph and some algebraic properties of a commutative ring. We prove that if AG(R) is a tree, then either AG(R) is a star graph or a path of… 

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References

SHOWING 1-10 OF 14 REFERENCES

On zero-divisor graphs of finite rings

The total graph and regular graph of a commutative ring

Zero-divisor semigroups and refinements of a star graph

The total graph of a commutative ring

The Annihilating-Ideal Graph of Commutative Rings I

Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$,

On bipartite zero-divisor graphs

On zero-divisor graphs of small finite commutative rings

The zero-divisor graph of a commutative semigroup

An undirected graph Γ(S) is associated to each commutative multiplicative semigroup S with 0. The vertices of the graph are labeled by the nonzero zero-divisors of S , and two vertices x,y are