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        "description": "Finite type as an algebra and finite as a module are different hypotheses; neither means finite cardinality. Zariski's field lemma, the Nullstellensatz and the principal-ideal theorem are acknowledged inherited results or applications, not separately claimed inventions here. Later fraction-field characterizations use the paper's convention that ideals exclude both 0 and R; no modern unrestricted minimal-prime claim is inferred. No noncommutative, non-Noetherian or arbitrary subalgebra finiteness claim.",
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      "statement": "# Artin–Tate: finite-type intermediate-ring lemma\n\nFor commutative rings R⊂T⊂S with unit and R Noetherian, if S is a finitely generated R-algebra and a finite T-module, then T is a finitely generated R-algebra. Artin and Tate prove this by extracting a finite coefficient subalgebra and using Noetherian module finiteness, and give a short proof of Zariski's field lemma as an application.\n\n## Scientific scope\n\nFinite type as an algebra and finite as a module are different hypotheses; neither means finite cardinality. Zariski's field lemma, the Nullstellensatz and the principal-ideal theorem are acknowledged inherited results or applications, not separately claimed inventions here. Later fraction-field characterizations use the paper's convention that ideals exclude both 0 and R; no modern unrestricted minimal-prime claim is inferred. No noncommutative, non-Noetherian or arbitrary subalgebra finiteness claim.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nEmil Artin; John T. Tate\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [A note on finite ring extensions (original1951)](https://www.jstage.jst.go.jp/article/jmath1948/3/1/3_1_74/_pdf)\n",
      "summary": "For commutative rings R⊂T⊂S with unit and R Noetherian, if S is a finitely generated R-algebra and a finite T-module, then T is a finitely generated R-algebra. Artin and Tate prove this by extracting a finite coefficient subalgebra and using Noetherian module finiteness, and give a short proof of Zariski's field lemma as an application.",
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    "id": "artin-tate-finite-type-intermediate-ring-1951",
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    "title": "Artin–Tate: finite-type intermediate-ring lemma",
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    "what_changed": "For commutative rings R⊂T⊂S with unit and R Noetherian, if S is a finitely generated R-algebra and a finite T-module, then T is a finitely generated R-algebra. Artin and Tate prove this by extracting a finite coefficient subalgebra and using Noetherian module finiteness, and give a short proof of Zariski's field lemma as an application.",
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