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The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Basing on the recent \"general bilinear forms\", we present a general machinery to attack the problem, and use it to show that (C)$\\iff$(B) when $R$ is semilocal or $\\mathbb{Q}$-fin"},"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":"1305.5139","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-05-22T13:56:23Z","cross_cats_sorted":[],"title_canon_sha256":"2be699ba9f19707a32b6bdd8e82fea63df4bb516b5dafe238de1be3dbbfe789f","abstract_canon_sha256":"9ced04829731b956366dc173e085561373a30e11f47104a1ad5b3a0928120921"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:36.449556Z","signature_b64":"F0uRni+R0NoABHzIC+khXtku43IlB/suxrxAF5jImWSIMR3xhhKRxFWwD/hhbdYY+gUI9qU8rGObq70dLxB4Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0c0bc2ae5a9fe50ea536d48cb1eb67e0ba948be81d3877fc3d2848e7d38fbca","last_reissued_at":"2026-05-18T02:19:36.449126Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:36.449126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rings That Are Morita Equivalent to Their Opposites","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Uriya A. First","submitted_at":"2013-05-22T13:56:23Z","abstract_excerpt":"We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an anti-automorphism, (C) $R$ is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. 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