The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

Abstract

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted 𝔸𝔾(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted 𝔸𝔾 _I (R). We discuss when 𝔸𝔾 _I (R) is bipartite. We also give some results on the subgraphs and the parameters of 𝔸𝔾 _I (R).

Key Words:

• Annihilating-ideal graph (with respect to an ideal)
• Bridge
• Clique number
• Cut-point
• Girth
• r-Partite graph

2010 Mathematics Subject Classification:

• 05C75
• 13A15
• 13A99

ACKNOWLEDGMENTS

The authors would like to thank the referee whose fruitful comments have improved this article.

The research of the second author was in part supported by grant no. 90160034 from IPM.

The research of the fourth author was in part supported by grant no. 90130035 from IPM.

Notes

Communicated by I. Swanson.