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# Questions tagged [noetherian]

For questions on Noetherian rings, Noetherian modules and related notions.

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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
0answers
13k views

### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
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271 views

### Exercise 5.5.M, Foundations of Algebraic Geometry by Ravi Vakil

In the following $A$ is a Noetherian ring. The basic definition here is: A prime $p\subsetneqq A$ is said to be associated to $M$ if $p$ is the annihilator of an element $m$ of $M$. The exercise ...
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126 views

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

Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one. Let $A$ be in $\mathsf{CRing}$. Question 1. ...
1answer
224 views

### Ideal in polynomial ring which contains no non-zero prime ideal

Let $J$ be a non-zero ideal in $\mathbb C[X,Y]$ such that $J$ contains no non-zero prime ideal. Then is it true that $J$ has height $1$ ? Possible approach: Since $\mathrm{ht}(J^n)=\mathrm{ht}(J)$ ...
0answers
125 views

### Integral domain over which every non-constant irreducible polynomial has degree 1

Let $R$ be an integral domain such that any polynomial $f(X) \in R[X]$ , which is irreducible in $R[X]$, has degree $1$. Then is it true that $R$ is a field ? If this is not true in general , What if ...
0answers
68 views

### Integral closure and $\bigcap \mathfrak{a}^n$

Let $R$ be a domain such that $\bigcap_{n=1}^\infty \mathfrak{a}^n=0$ holds for all proper ideals $\mathfrak{a}$ of $R$ (this holds, for example, if $R$ is Noetherian). Let $K$ be the quotient field ...
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43 views

### Non-Noetherian Domain Which is Locally Noetherian

Let $R$ be an integral domain such that the localization, $R_{\mathfrak p}$, at each prime ideal, $\mathfrak p \le R$ is Noetherian. Then is $R$ necessarily Noetherian? In the case of $R$ not ...
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126 views

### Integral domains $R$ such that any ring $R \subset S \subset K$ is Noetherian

Which integral domains $R$ with fraction field $K$ have the property that for any intermediate ring $R \subset S \subset K$, $S$ is Noetherian? I'm not really good at naming things, but let's call ...
0answers
495 views

### Ideals generated by regular sequences are generated by regular sequences in any order/Eisenbud, Exercise 17.6

I have a question regarding an exercise I found in Eisenbud's Commutative Algebra with a view towards Algebraic Geometry: Exercise 17.6: Any ideal of a Noetherian ring generated by a regular ...
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### Characterization of Noetherian module

I guess this is a simple question but I don't see the answer. Definition: $M$ is Noetherian if every chain of submodules stabilize. Theorem: $M$ is Noetherian module iff every nonempty set $S$ of ...