AWhat 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.
Scientific scope
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. 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.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. 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.
Contributors
- The named authors in the research record · SCIENCE: 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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