# Questions tagged [noetherian]

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

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### Is the decomposition of an Artinian or Noetherian module unique? [duplicate]

If $M$ is an Artinian or Noetherian module, then $M$ can be decomposed into the direct sum of a finite set of indecomposable submodules. Is the decomposition of $M$ unique?
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### Does there exist a more "quantitative" version of length for Noetherian or Artinian modules/rings?

I suppose this is a rather odd question, or at least maybe one more suited for MathOverflow, but I'll ask this here first as I'm more acquainted with MSE. In any case, this might be more "fuzzy&...
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### Find algebraically independent elements $z_1, \ldots, z_n$ such that $B = k[X, X^{-1}]$ is integral over $k[z_1, \ldots, z_n]$.

Let $k$ be an algebraically closed field. Find algebraically independent elements $z_1, \ldots, z_n$ such that $B = k[X, X^{-1}]$ is integral over $k[z_1, \ldots, z_n]$. We want to find a polynomial ...
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### Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.

Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$. The first thing I tried was to see that \displaystyle\bigcap_{n = 1}^\...
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### Possible inequality of krull dimension of local injection of Noetherian local domains

If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?
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### Converse to Composition of Finite Homomorphisms is Finite

Let $A,B,C$ be rings. I know if $A \to B \to C$ where $A \to B$ and $B \to C$ are finite, then $A \to C$ is finite (i.e. as modules). My question is, if we know $A \to C$ is finite, does it follow ...
• 1,498
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### Given an inverse sequence of functors determined on a subcategory, when is the limit determined on that subcategory?

I will first state the general version of my question, but I do have a specific context in mind in which second I'll dance around. (1.) Let $\mathsf{C}$ be a full subcategory of a category $\mathsf{D}$...
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1 vote
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### Show quotient of ideals is Noetherian

Not sure how to approach this problem: Let A be a commutative ring, I,J ideals of A such that A/I is a Noetherian ring. Show that J/(I ∩ J) is Noetherian as a A-module So far I only concluded that if ...
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### Chain of kernels of maps induced from picking a basis of modules

Let $R$ be a Noetherian ring. Suppose that $M_1$ is a finitely generated $R$-module, so that after picking a list of $n_1$ generators, $M_1$ is isomorphic to $R^{n_1}/M_2$, where $M_2$ is another ...
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### Is each component of a graded module over a $k$-algebra a finite-dimensional vector space?

I have some problems with an argument in a proof of a lemma: Let $M = \oplus_{-\infty}^{\infty} M_n$ be a finitely generated graded $A$-module and $A=\oplus_{n\geq 0} A_n$ a graded commutative ring ...
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### Proving that if $J=\bigcap_{n\geq0}I^n$ then $IJ = J$

I'm having some trouble with the following exercise: Let $R$ be a Noetherian commutative ring, $I$ and ideal and $J=\bigcap_{n\geq0}I^n$. Show that $IJ = J$. (Hint: Assume that $J\not\subseteq IJ$ ...
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1 vote