The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal |
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Authors: | F Aliniaeifard M Behboodi E Mehdi-Nezhad |
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Institution: | 1. Department of Mathematics , Brock University , St. Catharines , Ontario , Canada;2. Department of Mathematical Science , Isfahan University of Technology , Isfahan , Iran;3. School of Mathematics , Institute for Research in Fundamental Sciences (IPM) , Tehran , Iran;4. Department of Mathematics , University of Cape Town , Cape Town , South Africa |
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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). |
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Keywords: | Annihilating-ideal graph (with respect to an ideal) Bridge Clique number Cut-point Girth r-Partite graph |
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