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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 $3$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.08357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-12-26T10:14:34Z","cross_cats_sorted":[],"title_canon_sha256":"d379bb7eedae648da090fd62a402a908a4eca02f9cf9f8cf458a4891f744c631","abstract_canon_sha256":"fcb759dce7b4e0bc1a3c05f1e254558c3f44253566263f01410b35d0e1228b09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:56.838982Z","signature_b64":"54mtvdX3+4Yi0Eyye3NgGyJYAzezsYn2NtCo3BBjeRAm9gq/MRLuuHWPVjvO1tFiiRIs0UHaNTMqrojlG40MAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caa016d9f826823b69ec96db1ea7d69d89cc900843d2b8f39c54c3917f0dd0cd","last_reissued_at":"2026-05-18T00:53:56.838546Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:56.838546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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