# Minimal prime ideals and cycles in annihilating-ideal graphs

@article{Aalipour2013MinimalPI,
title={Minimal prime ideals and cycles in annihilating-ideal graphs},
journal={Rocky Mountain Journal of Mathematics},
year={2013},
volume={43},
pages={1415-1425}
}
• Published 1 October 2013
• Mathematics
• Rocky Mountain Journal of Mathematics
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 13 REFERENCES
The Annihilating-Ideal Graph of Commutative Rings I
• Mathematics
• 2008
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
• Mathematics
Discret. Math.
• 2009
The zero-divisor graph of a commutative semigroup
• Mathematics
• 2002
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
Introduction to commutative algebra
• Mathematics
• 1969
* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *