# Questions tagged [noetherian]

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

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### $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
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### Artinian - Noetherian rings and modules suggest study guide

What text or any document that has gathered this part of Algebra theory. Thanks. Pd: I seek on variety's book of commutative algebra but the subject is partially dealt
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### Question about proof of Krull principal ideal theorem

How can we explain the following step in the proof of Krull principal ideal theorem: $l\{ ((z):x^n)/(z) \}$ or $l\{ ((x^n):z)/(x^n) \}$ is finite? $l(M)$ - length of module.
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### About a short proof of Krull principal ideal theorem

How from this theorem I can get a proof of Krull principal ideal theorem? I understand that w.l.g. we can prove it for a Noetherian local ring. But why we can consider that $(x)$ is $M$-primary?
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### If $A$ is a commutative, unitary ring and $I$ an ideal of $A$ such that $I$ and $A/I$ are Noetherian rings, then $A$ is Noetherian?

If $A$ is a commutative, unitary ring and $I$ an ideal of $A$ such that $I$ and $A/I$ are Noetherian rings, then $A$ is Noetherian ? I know just that if $A$ is Noetherian then $A/I$ is Noetherian....
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### Noether's normalization lemma in practice (example)

I would like to know how to use the Noether's normalization lemma in practice. Noether's normalization lemma Let $k$ an infinite field, and $k[a_1,\dots ,a_n]$ be a finite $k$-algebra. There ...
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### Every non-Noetherian module has a submodule maximal with respect to being not finitely generated. [duplicate]

Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated whenever $N<A\leq M$. The question is related to If $M$ isn't ...
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### Unusual proof that $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian

Let $M$ be an $R$-module for a ring $R$. Let $N$ be a submodule of $M$. I read that one can prove that $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian using these two results: 1) If $M$ ...
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### Can one add reasonable assumptions that we have $depth\ R \geq depth\ M$ for every $R$-module $M$?

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. Here, Hanno shows that one can have any inequality between $depth\ R$ and $depth\ M.$ Still, the ...
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### Uncountably many left ideals?

Let $R$ be a following subring of $M_2(\mathbb{C}):$ \begin{equation*} R = \left\{ \begin{bmatrix} a & r \\ 0 & s \end{bmatrix} ~:~ a\in \mathbb{Q} ~\mbox{...
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### ACC on chains of finitely generated submodules

Noetherian rings are those having ascending chain condition on ideals. There is also literature concerning ACC on n-generated (i.e., generated by n elements) ideals; see e.g, Commutative Rings with ...
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### Verify proof that if $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian

I have to prove that If $M,N$ are $R$-modules and $M$ is Noetherian, $N$ is finitely generated, then $M\otimes_R N$ is Noetherian We let $S$ be a non-finitely generated submodule of $M\otimes_R M$....
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### Proper ideal $I \implies \exists$ prime ideals $P_i$ such that $P_1 \cdots P_n \subset I$.

Let the below ideals be in a commutative Noetherian ring $R$. Corollary 22. (3) There are prime ideals $P_1, \dots, P_n$ (not necc. distinct) $\supset I$ such that $P_1\cdots P_n \subset I$. (Out ...
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### If $R$ is a noetherian ring, then minimal primes of $R[x]$ are exactly the ideals $P[x]$ of $R[x]$ where $P$ is a minimal prime of $R$

Show that if $R$ is a noetherian ring, then minimal primes of $R[x]$ are exactly the ideals $P[x]$ of $R[x]$ where $P$ is a minimal prime of $R$. Definition of a minimal prime ideal of a ring $R$: ...
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### Normal Noetherian rings of dimension at least 1

We want to pick $I_g$ to be "maximal" but what is the partial ordering to which it is maximal? For two $f,g \in A' - A$, I don't see how $I_f$ and $I_g$ would be related via subsets. How can we see ...
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### Equivalence of definitions of Noetherian Ring, another proof.

Let $R$ be a commutative ring with unity, then the following are equivalent -1. Every ideal in $R$ is finitely generated -3. Every nonempty collection of ideals of $R$ has a maximal element I will ...
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### Intersection of height one prime ideals

Let $A$ be a commutative, integrally closed, noetherian ring and as $\mathfrak p$ ranges over height one prime ideals, we have: $$A = \bigcap_\mathfrak p A_{\mathfrak p}.$$ The proof I have seen ...
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### matrix ring Noetherian, Artinian, semisimple?

Let $k$ be a field and $\Lambda=\begin{bmatrix} k & 0 \\ k^2 & k[x]/(x^2) \end{bmatrix}$. This ring is an algebra over $k$. (a) What is $\dim_k \Lambda$? (b) Is $\Lambda$ a left ...
Let the $R=\Big\{\begin{pmatrix}a & 0 \\ b & c \end{pmatrix} : a,b,c \in \mathbb{R}\Big\}$ be a ring. Is it: 1) Artinian? 2) Noetherian? If a ring $R$ is noetherian (artinian), then every ...
### Noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$
Is there a noetherian local domain $A$ with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$? This is a follow up question to: Does codimension behave weirdly even in local rings?