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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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
Emil Artin; John T. Tate
Source-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.
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English catalogue narrative
Artin–Tate: finite-type intermediate-ring lemma
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.
Scientific scope
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.
Reading status
This 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.
People and systems
Emil Artin; John T. Tate
Source-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.