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Let $R$ be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors $R^\\bullet$ can be written as a product of atoms. We show that, if $R$ is bounded and every stably free right $R$-ideal is free, then there exists a transfer homomorphism from $R^{\\bullet}$ to the monoid $B$ o"},"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":"1605.09274","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-05-30T15:22:59Z","cross_cats_sorted":[],"title_canon_sha256":"933c5e05d7364966011ce4b74b62ffb837ed8d683951dcbdf8e9f81d9ad70e2c","abstract_canon_sha256":"100d842af9895fd2559428d19841851ec5d6708f620f6e94c892530c42206a8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:06.517582Z","signature_b64":"m5O6Wzn6yXcA4/r3aajrjH4rLJ628wK7YZbmCfYcxwucYDhQ4sLAwzdq0/sNXcgI1QDdbdnzcry5K3VvBgAPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd00c0e7afecf3b753a54871fb9d60abcea4ad1f592df3cee6130b19636bf74a","last_reissued_at":"2026-05-17T23:49:06.516980Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:06.516980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Factorizations in bounded hereditary Noetherian prime rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Daniel Smertnig","submitted_at":"2016-05-30T15:22:59Z","abstract_excerpt":"If $H$ is a monoid and $a=u_1 \\cdots u_k \\in H$ with atoms (irreducible elements) $u_1, \\ldots, u_k$, then $k$ is a length of $a$, the set of lengths of $a$ is denoted by $\\mathsf L(a)$, and $\\mathcal L(H)=\\{\\,\\mathsf L (a) \\mid a \\in H \\,\\}$ is the system of sets of lengths of $H$. Let $R$ be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors $R^\\bullet$ can be written as a product of atoms. We show that, if $R$ is bounded and every stably free right $R$-ideal is free, then there exists a transfer homomorphism from $R^{\\bullet}$ to the monoid $B$ o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.09274","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.09274","created_at":"2026-05-17T23:49:06.517083+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.09274v2","created_at":"2026-05-17T23:49:06.517083+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.09274","created_at":"2026-05-17T23:49:06.517083+00:00"},{"alias_kind":"pith_short_12","alias_value":"3UAMBZ5P5TZ3","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"3UAMBZ5P5TZ3OU5F","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"3UAMBZ5P","created_at":"2026-05-18T12:29:58.707656+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP","json":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP.json","graph_json":"https://pith.science/api/pith-number/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/graph.json","events_json":"https://pith.science/api/pith-number/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/events.json","paper":"https://pith.science/paper/3UAMBZ5P"},"agent_actions":{"view_html":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP","download_json":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP.json","view_paper":"https://pith.science/paper/3UAMBZ5P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.09274&json=true","fetch_graph":"https://pith.science/api/pith-number/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/graph.json","fetch_events":"https://pith.science/api/pith-number/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/action/storage_attestation","attest_author":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/action/author_attestation","sign_citation":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/action/citation_signature","submit_replication":"https://pith.science/pith/3UAMBZ5P5TZ3OU5FJBY7XHLAVP/action/replication_record"}},"created_at":"2026-05-17T23:49:06.517083+00:00","updated_at":"2026-05-17T23:49:06.517083+00:00"}