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We say that $A\\in M_n(R)$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map $\\{\\cdot, \\cdot\\}$ : $M_n(R)\\times M_n(R)\\to X$ the following two conditions are equivalent: (i) there exists a fixed element $w\\in X$ such that $\\{x,y\\}=w$ whenever $x\\circ y=A$, $x,y\\in M_n(R)$; (ii) there exists an $R$-linear map $T:M_n(R)^2\\to X$ such that $\\{x,y\\}=T(x\\circ y)$ for all $x,y\\in M_n(R)$. In this paper, we mainly prove that all the matrix uni"},"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":"1111.4108","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.OA","submitted_at":"2011-11-17T14:23:22Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"0873698ea043940741c504e6cba7b9e5d02bbda3c700a3acf43f3c5a7407e3e2","abstract_canon_sha256":"e6daa9084506d5de558ca8ec430e5a554fe473c1c5aaf1439fdc19a098f5e18a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:05.670554Z","signature_b64":"kKRJbN4LEZzW0p3odemnIbzhnzOGU/G+3F+uT9txHjddsEHZBVAk/zv42KnjNuEaIhKSN4Yl69a+c9f8gnutAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"320370ed3719292652245e94189273cf9bb791db90ed80c760e0d8cdd658a8c7","last_reissued_at":"2026-05-18T04:08:05.670024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:05.670024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Jordan product determined points in matrix algebras","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.OA","authors_text":"Yang Wenlei, Zhu Jun","submitted_at":"2011-11-17T14:23:22Z","abstract_excerpt":"Let $M_n(R)$ be the algebra of all $n\\times n$ matrices over a unital commutative ring $R$ with 6 invertible. We say that $A\\in M_n(R)$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map $\\{\\cdot, \\cdot\\}$ : $M_n(R)\\times M_n(R)\\to X$ the following two conditions are equivalent: (i) there exists a fixed element $w\\in X$ such that $\\{x,y\\}=w$ whenever $x\\circ y=A$, $x,y\\in M_n(R)$; (ii) there exists an $R$-linear map $T:M_n(R)^2\\to X$ such that $\\{x,y\\}=T(x\\circ y)$ for all $x,y\\in M_n(R)$. 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