Under what conditions is the tensor product of two non-commutative Noetherian domains also a Noetherian domain? To be more precise about the problem, I am looking at rings of fractions on such tensor products, as a method of generalising the coaction of a Hopf algebra $H$ on an algebra $A$. Thus the coaction on an element of $A$ may lie in a ring of fractions on $A\otimes H$ rather than just $A\otimes H$. Any useful information on this would be gratefully appreciated!
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$\begingroup$ Tensor product over what? $\endgroup$– abxCommented Jan 18, 2018 at 17:40
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$\begingroup$ For standard coaction of Hopf algebras it would just be the field. $\endgroup$– Edwin BeggsCommented Jan 18, 2018 at 21:08
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$\begingroup$ If $K$ is algebraically closed then a Noetherian domain is always of the form $K[X]$ for some irreducible affine variety $X$. The tensor product $K[X]\otimes_K K[Y]$ is isomorphic with $K[X\times Y]$, which is again a Neotherian domain. The only problem which might arise is when $K$ is not algebraically closed: for example if $L$ is a finite extension of $K$ then $L\otimes_K L$ is not a domain. Is the ground field algebraically closed in your case? $\endgroup$– Ehud MeirCommented Jan 19, 2018 at 10:37
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$\begingroup$ Apologies - I should say that for Hopf algebras the interesting case is where the algebras are non-commutative. $\endgroup$– Edwin BeggsCommented Jan 19, 2018 at 19:00
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6$\begingroup$ @Ehud Meir: for you, $K[[T]]$ is not a Noetherian domain? $\endgroup$– abxCommented Jan 20, 2018 at 10:27
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