Noetherian algebras over algebraically closed fields
Copyright policy )Noetherian algebras over algebraically closed fields
Let $k$ be an uncountable algebraically closed field and let $A$ be a countably generated left Noetherian $k$-algebra. Then we show that $A \otimes_k K$ is left Noetherian for any field extension $K$ of $k$. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over $k$ are finitely generated extensions of $k$. We give examples which show that $A\otimes_k K$ need not remain left Noetherian if the hypotheses are weakened.
10 pages
quotient division algebras, Strongly Noetherian, Algebra and Number Theory, Field extensions, Noetherian domains, Noetherian rings and modules (associative rings and algebras), Base change, Noetherian algebras, Mathematics - Rings and Algebras, Localization and associative Noetherian rings, strongly Noetherian algebras, Rings and Algebras (math.RA), Stably Noetherian, stably Noetherian algebras, FOS: Mathematics, finitely generated field extensions, Finite rings and finite-dimensional associative algebras, Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting), base change
quotient division algebras, Strongly Noetherian, Algebra and Number Theory, Field extensions, Noetherian domains, Noetherian rings and modules (associative rings and algebras), Base change, Noetherian algebras, Mathematics - Rings and Algebras, Localization and associative Noetherian rings, strongly Noetherian algebras, Rings and Algebras (math.RA), Stably Noetherian, stably Noetherian algebras, FOS: Mathematics, finitely generated field extensions, Finite rings and finite-dimensional associative algebras, Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting), base change
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