{"paper":{"title":"Weak clean index of a ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Dhiren Kumar Basnet, Jayanta Bhattacharyya","submitted_at":"2016-12-26T10:14:34Z","abstract_excerpt":"Motivated by the concept of clean index of rings, we introduce the concept of weak clean index of rings. For any element $a$ of a ring $R$ with unity, we define $ \\chi(a)=\\{e\\in R\\mid e^2=e\\text{ and }a-e \\mbox{ or } a+e \\mbox{ is a unit}\\}$. The weak clean index of $R$ is defined as $\\sup \\{|\\chi(a)|: a\\in R\\}$ and it is denoted by $\\win(R),$ where $| \\chi(a)| $ denotes the cardinality of the set $\\chi(a)$. In this article, we characterize rings of weak clean indices $1$, $2$ and $3$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08357","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"}