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This article describes some situations where $\\alpha$ is injective and some where it is not.\n  We prove that if $A$ is a right Ore localization of $R[a]$ and $B$ is a right Ore localization of $R[b]$, then $\\alpha$ is injective. For example, the group ring over $R$ of the free group on $\\{1+a,"},"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":"1211.6323","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-11-27T15:10:47Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2e4fe2057c7c592ddea7fc6022223ce798b68f316677ed31db244f104058fed4","abstract_canon_sha256":"689fceff01e255e7fc92c725aa381d23567717f0a00770ae31a9f0fea7d3d6ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:21.578297Z","signature_b64":"pv25Y+igbBdYqO8WxyC7BtakFgS5yb5Loo5k89pwaEgxTkHiszd+gj6VPLFhjAartV0TdqUH4wY9kLFlqN2ADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c9693962024cc6bfc7c73abd580c4e04c9a1eaf9586c40d3b0e472dfe954e78","last_reissued_at":"2026-05-18T02:16:21.577630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:21.577630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ring coproducts embedded in power-series rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Pere Ara, Warren Dicks","submitted_at":"2012-11-27T15:10:47Z","abstract_excerpt":"Let $R$ be a ring (associative, with 1), and let $R<< a,b>>$ denote the power-series $R$-ring in two non-commuting, $R$-centralizing variables, $a$ and $b$. Let $A$ be an $R$-subring of $R<< a>>$ and $B$ be an $R$-subring of $R<< b>>$, and let $\\alpha$ denote the natural map $A \\amalg_R B \\to R<< a,b>>$. This article describes some situations where $\\alpha$ is injective and some where it is not.\n  We prove that if $A$ is a right Ore localization of $R[a]$ and $B$ is a right Ore localization of $R[b]$, then $\\alpha$ is injective. 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