# Questions tagged [noetherian]

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16 questions
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### Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do ...
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### Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ?

Let $P$ be a non-finitely generated projective module over a commutative Noetherian ring. Is every finitely generated submodule of $P$ contained in some finitely generated direct summand of $P$ ? Or ...
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### Factorially closed, finitely generated $k$-sub-algebra $A$ of $k[X_1,…,X_n]$, where $n>3$, $k$ is algebraically closed of char $0$, $trdeg_k A=n-1$

Let $S$ be a sub-ring of a commuttaive ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S \implies a,b \in S$. My question is : Let $k$ be an algebraically ...
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### Can the Artin-Rees lemma be derived from Krull Intersection theorem?

The Krull Intersection theorem states that : For a finitely generated module $M$ over a Noetherian ring $R$ and any ideal $I$ of $R$, we have $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
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### Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field

If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ? If $R$ is normal (integrally ...
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### an easy example of valuation ring which is not noetherian？ [duplicate]

Is there an easy example of valuation ring which is not noetherian？
Does the category of noetherian commutative rings have pushouts? Background: If $X/S$ is an abelian scheme, then the relative Picard functor $\mathrm{Pic}_{X/S}$ is only defined on the category of ...