{"paper":{"title":"On graphs related to co-maximal ideals of a commutative ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Dancheng Lu, Houyi Yu, Meng Ye, Tongsuo Wu","submitted_at":"2011-06-01T01:16:12Z","abstract_excerpt":"This paper studies the co-maximal graph $\\Om(R)$, the induced subgraph $\\G(R)$ of $\\Om(R)$ whose vertex set is $R\\setminus (U(R)\\cup J(R))$ and a retract $\\G_r(R)$ of $\\G(R)$, where $R$ is a commutative ring. We show that the core of $\\G(R)$ is a union of triangles and rectangles, while a vertex in $\\G(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\\G(R)$ are identical with the number of maximal ideals of $R$. A graph $\\G_r(R)$ is also introduced on the vertex set $\\{Rx|\\,x\\in R\\setminus (U(R)\\cup J(R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0072","kind":"arxiv","version":1},"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"}