# Questions tagged [noetherian]

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

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### If the source of a morphism is a noetherian scheme, the morphism is quasi-compact.

I was wondering it while studying Hartshorne. In II.4.8 proof of the corollary (This part is about valuative criterion of properness.), it is written that a morphism from noetherian scheme is ...
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### Is $\oplus_{\mathbb{N}}\mathbb{Z}_2$ Artinian but not Noetherian?

Just as in the title: I've seen the statement that Artinian rings are Noetherian several times (eg Commutative artinian ring is noetherian) but if we take $R=\oplus_\mathbb{N}\mathbb{Z}_2$, it seems ...
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### direct sum of noetherian modules is noetherian

We say that an $A$-module $M$ is Noetherian if all of its submodules are finitely generated. Having that definition in mind can anyone give me some hints to prove that if $M$ and $N$ are Noetherian ...
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### $M$ is a $\Bbb Z/36 \Bbb Z$-module and $\bar 6 \cdot M = \{0\}$. Prove: $M$ is Noetherian $\iff$ it's Artinian

Suppose that $M$ is a $\Bbb Z/36 \Bbb Z$-module such that $\bar 6 \cdot M = \{0\}$. Prove that $M$ is Noetherian $\iff$ $M$ is Artinian. I managed to prove the forward direction: if $M$ is Noetherian,...
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### Are Noetherian Rings Goldie Rings?

We know that a ring $R$ is said to be Right-Goldie if $R$, as a right $R$ module, satisfies: (i) $R$ has finite uniform dimension; (ii) every ascending chain of right annihilators of $R$ terminate. ...
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### If every ideal I in R is contained in a finite series of ascending ideals, prove that only finitely many ideals of any kind contain I.

If a Noetherian ring is defined by the fact that all ideals are contained within a finite series of ascending ideals, how does this prove that the initial ideal is contained within finitely many ...
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### If $I$ is finitely generated nilpotent and $R/I^{n-1}$ is noetherian then $R$ is noetherian

If $I$ is a finitely generated ideal of a commutative ring $R$ with $1$ such that $I^n = \{0\}$ and $R/I^{n-1}$ is noetherian, then $R$ is also noetherian. I don't know what I should do. If I can ...
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### If $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$

Let $R$ be a commutative unitary ring and $I,J$ be two ideals of $R$. I need to prove that if $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$ It seems that a direct ...
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### System of parameters in Noetherian local rings

I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says: Let $R$ be a Noetherian local ring with maximal ideal $m$. Then there exists an $m$-primary ideal ...
### The product $JR$ of a proper ideal $J$ and the nontrivial ideal $R$
There is a proposition on page 390 of Hungerford's algebra'' (1974) as follows: Proposition 4.6. Let $J$ be an ideal in a commutative ring $R$ with identity. Then $J$ is contained in every maximal ...
### $R$ 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 ?