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This description clarifies Auslander-Gruson-Jensen duality and also the duality between definable subcategories of right A-modules and those of left A-modules."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The existence of this duality arises from the fact that mod-(mod-A) is the free abelian category over the pre-additive category A with a single object."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Gives a simple description of the free abelian category to clarify Auslander-Gruson-Jensen duality between definable subcategories of right and left modules over a ring."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A simple description of the free abelian category clarifies Auslander-Gruson-Jensen duality for modules over a ring."}],"snapshot_sha256":"3786924c8e07e851dd59c45e7a2555d8b39154bcbbca3275e3d8636e321bf4f1"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"4fb821ebb15672760066458e86112a1267191637d4c42791e3154a7ebdfb6342"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T15:23:21.859858Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.03458/integrity.json","findings":[],"snapshot_sha256":"58d0578624a76cf4de0627a4370136cd26d5169454081bb64488356acc467bd4","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\\rm mod\\text{-}(mod\\text{-}A)\\rightarrow mod\\text{-}(mod\\text{-}A^{op})$. The existence of this duality arises from the fact that $\\rm mod\\text{-}(mod\\text{-}A)$ is the free abelian category over the pre-additive category $A$ with a single object.\n  In this note, first, we give a simple description of the free abelian category. 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