{"paper":{"title":"On the structure of quaternion rings over $\\mathbb{Z}/n \\mathbb{Z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Antonio Oller-Marcen, Celino Miquel, Jose Maria Grau","submitted_at":"2014-02-05T07:20:55Z","abstract_excerpt":"In this paper we investigate the structure of $\\left(\\frac{a,b}{\\Z{n}}\\right)$, the quaternion rings over $\\Z{n}$. It is proved that these rings are isomorphic to $\\left(\\frac{-1,-1}{\\Z{n}}\\right)$ if $ a \\equiv b\\equiv -1 \\pmod{4}$ or to $\\left(\\frac{1,1}{\\Z{n}}\\right)$ otherwise. We also prove that the ring $\\left(\\frac{a,b}{\\Z{n}}\\right)$ is isomorphic to $\\mathbb{M}_2(\\Z{n})$ if and only if $n$ is odd and that all quaternion algebras defined over $\\Z{n}$ are isomorphic if and only if $n \\not \\equiv 0 \\pmod{4}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0956","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"}