# 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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