# Questions tagged [noetherian]

The tag has no usage guidance.

43 questions
Filter by
Sorted by
Tagged with
734 views

### Are topological PID's Noetherian?

Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...
730 views

### Excellent property of rings

Let $A$ be a commutative ring. If $A$ is an excellent ring, is the reduced ring $A/\sqrt{(0)}$ also an excellent ring?
164 views

1 vote
110 views

81 views

219 views

### Artin-Rees lemma for multiplicative subsets?

The classical Artin-Rees lemma tells the following. Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Let $M$ be a finitely generated $R$-module and $N\subset M$ be a submodule. ...
1 vote
186 views

### 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 ...
527 views

### 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 ...
773 views

### 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 ...
438 views

1 vote
76 views

### 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 $R$ ...
1 vote
291 views

Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$. Then is the natural map $j:M \to M^{**}$ defined as $j(m)(... 2 votes 1 answer 400 views ### Finitely generated submodule of non-finitely generated projective module is contained in some proper direct summand ? Let$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$? Or ... 5 votes 0 answers 324 views ### Can the Artin-Rees lemma be derived from Krull Intersection theorem? The Krull Intersection theorem states that : For a finitely generated module$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$. ... 1 vote 1 answer 207 views ### Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field If$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$is normal (integrally ... 4 votes 1 answer 360 views ### torsion free modules$M$over Noetherian domain of dimension$1$for which$l(M/aM) \le (\dim_K K \otimes_R M) \cdot l(R/aR), \forall 0 \ne a \in R$Let$R$be a Noetherian domain of Krull-dimension$1$(i.e. every non-zero prime ideal maximal). Let$M$be a torsion free$R$-module . Let$K$be the fraction-field of$R$and let$r=\dim_K S^{-1}M=\... 964 views

### 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 ? ... 276 views

### 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 ... 372 views

### 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. ...
183 views

4k views

### 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 ...