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Let $T\\mathop{\\mathrm{grmod}}$ denote the category of finite graded left $T$-modules, and $M\\in T\\mathop{\\mathrm{grmod}}$. Then the following are equivalent: (1) $\\hat M$ is an indecomposable $\\hat T$-module, where $\\widehat{(-)}$ denotes the $\\mathfrak{m}$-adic completion; (2) $M_{\\mathfrak{m}}$ "},"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":"2306.14523","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2023-06-26T08:56:37Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"b984a288523a69b1846f11a76bbfc1add2ef2c00e983b04431b39cc2554054e6","abstract_canon_sha256":"fffafac8b8af5f23e4d55cb3e938d74310ead1d46445b38cb0351c9495d678d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:24:37.450444Z","signature_b64":"l6JhNoJHE93Y7DtFcC+WogsjtGzZugzR7g70hcTBtETYbdKU1y9Em612qcb1Zn3ACtc+mkOqIWWuEWSwcji1AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"699823fc9c5dcda4ad0020cf367a5e9757c6b72b0bfeb6da136fe1a07639fd11","last_reissued_at":"2026-07-05T06:24:37.449942Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:24:37.449942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Indecomposability of graded modules over a graded ring","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Mitsuyasu Hashimoto, Yuntian Yang","submitted_at":"2023-06-26T08:56:37Z","abstract_excerpt":"Let $R=\\bigoplus_{i\\geq 0}R_i$ be a Noetherian commutative non-negatively graded ring such that $(R_0,\\mathfrak{m}_0)$ is a Henselian local ring. Let $\\mathfrak{m}$ be its unique graded maximal ideal $\\mathfrak{m}_0+\\bigoplus_{i>0}R_i$. Let $T$ be a module-finite (non-commutative) graded $R$-algebra. Let $T\\mathop{\\mathrm{grmod}}$ denote the category of finite graded left $T$-modules, and $M\\in T\\mathop{\\mathrm{grmod}}$. 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