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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 ","authors_text":"Alexandr Grishkov, Marina Rasskazova, Petr Vojt\\v{e}chovsk\\'y","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-18T16:58:08Z","title":"Automorphic loops arising from module endomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06521","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:50211251437bb035b574b49c9b6c53da3ef3603b196d6b6ca14dace5edbbf580","target":"record","created_at":"2026-05-18T00:27:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"81002f1abd1d9125b1a6c1e2da9d65ec7e67cfbcbf9819db207f9b9772da7881","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-18T16:58:08Z","title_canon_sha256":"51f96644194c22d684f4c02587a05279688a8de8e69d5010841a4cad16a55ca3"},"schema_version":"1.0","source":{"id":"1712.06521","kind":"arxiv","version":1}},"canonical_sha256":"025d47d951f59112f1bb22865089bbe7cf7c605a6dbcf51c6d4a797bdd51af5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"025d47d951f59112f1bb22865089bbe7cf7c605a6dbcf51c6d4a797bdd51af5a","first_computed_at":"2026-05-18T00:27:47.639251Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:47.639251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lOvjw2WJP4ZAr+HF2wIU0G2qqDB9laFPHF+W80MVWCOVCRtsTBlS/rPQ/z8zgqPRr+A9NqKagfMYQ+nNxXvFBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:47.639681Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.06521","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50211251437bb035b574b49c9b6c53da3ef3603b196d6b6ca14dace5edbbf580","sha256:8d7cd23bd17586d9fa3a5679e1e0b115769b5607e94931445b43a33fab1fde22"],"state_sha256":"b367f135764808a9240ebc50dc44757ab900427a5af8590094f091566e209d25"}