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Let $R$ be a commutative ring, $V$ an $R$-module, $E=\\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\\in W$ and $1+a$ is invertible for every $a\\in W$. Then $Q_{R,V}(W)$ defined on $W\\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an automorphic loop.\n  A special case occurs when $R=k<K=V$ is a field extension and $W$ is a $k$-subspace of $K$ such that $k1\\cap W = 0$, naturally embedded into "},"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":"1712.06521","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-18T16:58:08Z","cross_cats_sorted":[],"title_canon_sha256":"51f96644194c22d684f4c02587a05279688a8de8e69d5010841a4cad16a55ca3","abstract_canon_sha256":"81002f1abd1d9125b1a6c1e2da9d65ec7e67cfbcbf9819db207f9b9772da7881"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:47.639681Z","signature_b64":"lOvjw2WJP4ZAr+HF2wIU0G2qqDB9laFPHF+W80MVWCOVCRtsTBlS/rPQ/z8zgqPRr+A9NqKagfMYQ+nNxXvFBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"025d47d951f59112f1bb22865089bbe7cf7c605a6dbcf51c6d4a797bdd51af5a","last_reissued_at":"2026-05-18T00:27:47.639251Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:47.639251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphic loops arising from module endomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexandr Grishkov, Marina Rasskazova, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2017-12-18T16:58:08Z","abstract_excerpt":"A loop is automorphic if all its inner mappings are automorphisms. We construct a large family of automorphic loops as follows. Let $R$ be a commutative ring, $V$ an $R$-module, $E=\\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\\in W$ and $1+a$ is invertible for every $a\\in W$. Then $Q_{R,V}(W)$ defined on $W\\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an automorphic loop.\n  A special case occurs when $R=k<K=V$ is a field extension and $W$ is a $k$-subspace of $K$ such that $k1\\cap W = 0$, naturally embedded into "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06521","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.06521","created_at":"2026-05-18T00:27:47.639331+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.06521v1","created_at":"2026-05-18T00:27:47.639331+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.06521","created_at":"2026-05-18T00:27:47.639331+00:00"},{"alias_kind":"pith_short_12","alias_value":"AJOUPWKR6WIR","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AJOUPWKR6WIRF4N3","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AJOUPWKR","created_at":"2026-05-18T12:31:05.417338+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/AJOUPWKR6WIRF4N3EKDFBCN347","json":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347.json","graph_json":"https://pith.science/api/pith-number/AJOUPWKR6WIRF4N3EKDFBCN347/graph.json","events_json":"https://pith.science/api/pith-number/AJOUPWKR6WIRF4N3EKDFBCN347/events.json","paper":"https://pith.science/paper/AJOUPWKR"},"agent_actions":{"view_html":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347","download_json":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347.json","view_paper":"https://pith.science/paper/AJOUPWKR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.06521&json=true","fetch_graph":"https://pith.science/api/pith-number/AJOUPWKR6WIRF4N3EKDFBCN347/graph.json","fetch_events":"https://pith.science/api/pith-number/AJOUPWKR6WIRF4N3EKDFBCN347/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347/action/storage_attestation","attest_author":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347/action/author_attestation","sign_citation":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347/action/citation_signature","submit_replication":"https://pith.science/pith/AJOUPWKR6WIRF4N3EKDFBCN347/action/replication_record"}},"created_at":"2026-05-18T00:27:47.639331+00:00","updated_at":"2026-05-18T00:27:47.639331+00:00"}