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Results tagged with noetherian
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user 461
9
votes
1
answer
818
views
Are epimorphic endomorphisms of noetherian commutative rings always injective?
Eric Wofsey gave an example of an epimorphic endomorphism of a noetherian ring which is not surjective. … (It is well known, and easy to prove, that surjective endomorphisms of noetherian rings are automatically bijective.) …
- 4,416
13
votes
1
answer
465
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Does the Cantor-Schröder-Bernstein Theorem hold in the category opposite to the category of ...
(3) If $f:A\to B$ and $g:B\to A$ are surjective morphisms of noetherian rings, are $A$ and $B$ isomorphic? The answer is Yes, because surjective endomorphisms of noetherian rings are isomorphisms. … But epimorphic endomorphisms of noetherian rings are not always isomorphisms: see this answer of Eric Wofsey. …
- 4,416
27
votes
2
answers
2k
views
Is every commutative ring a limit of noetherian rings?
I don't care that much about Question 4, but I'm very curious about Question 5, which is
Do binary coproducts always exist in the category of noetherian commutative rings?
End of edit. … Is $\mathbb Z[x_1,x_2,\dots]$ a limit of noetherian rings?
(The $x_i$ are indeterminates.)
Question 5. Do binary coproducts exist in $\mathsf{Noeth}$? …
- 4,416
2
votes
1
answer
249
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Are unique prime ideal factorization domains locally noetherian?
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Does it follow that $A$ is locally noetherian? … Here are a couple of comments:
In this answer Badam Baplan pointed out that locally noetherian domains do have the indicated property, and that that some non-noetherian domains are locally noetherian …
- 4,416