{"paper":{"title":"The Annihilating-Ideal Graph of Commutative Rings I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Mahmood Behboodi, Zahra Rakeei","submitted_at":"2008-08-23T13:14:25Z","abstract_excerpt":"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 $R$, denoted by ${\\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\\Bbb{A}}(R)^*:={\\Bbb{A}}(R)\\setminus\\{(0)\\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.3187","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}