# Questions tagged [noetherian]

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### Creating prime ideals in rings

Are there different ways to create prime ideals in a ring other than taking quotients? I recently came across a construction of a prime ideal in a Noetherian ring $A$ given in the book on Algebraic ...
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### On the annihilator of a module

Question. Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Does there always exist an element $s\in M$ such that $\mathrm{Ann}(s)=\mathrm{Ann}(M)$? Remark. The annihilator of a ...
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### Are epimorphic endomorphisms of noetherian commutative rings always injective?

This question was asked, but not answered, on Mathematics Stackexchange. [In this post "ring" means "commutative ring with one".] Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism ...
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### Localizing prime ideals over Noetherian rings

Let $R$ be a prime Noetherian ring which is not necessarily commutative. Consider the two natural ways to "extend" $R$: $R[x]$ and $M_n(R)$, polynomials over $R$ and $n$ by $n$ matrices over $R$ ...
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### When does prime elements remain prime in certain integral extension

Let $R$ be an integral domain and $\bar R$ denote its integral closure in the fraction field (i.e. normalization). If $p\in R$ is a prime element in $R$, then does $p$ remain prime in $\bar R$ also ? ...
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### Noetherian rings as homomorphic image of finite direct product of Noetherian domains?

A theorem of Hungerford says that : Every PIR (principal ideal ring , obviously commutative ) is a homomorphic image of a finite direct product of PID s . My question is , is there a similar criteria ...
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### Are finitely generated objects in a locally Noetherian category also finitely presented?

Let $\mathcal C$ be a locally Noetherian category. Then, we know that every finitely generated object is Noetherian. I cannot seem to be able to prove that Noetherian objects are finitely presented. ...
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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 ...