# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### Noetherian local domain with a prime $P$ so that $\operatorname{ht}P+\dim A/P<\dim A$?

Is there a noetherian local domain 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?
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### Free commutative ring functor

The free commutative ring on a set $X$ is the polynomial ring with variables the elements of $X$. This polynomial ring is the free (additive) abelian group on the free (multiplicative) abelian monoid ...
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### In stacks project: Polynomial ring over UFD is UFD

Lemma 10.119.8 in this page on Stacks project states that a polynomial ring over a UFD is a UFD. It uses Nagata's criterion for factoriality (10.119.7): If $A$ is a domain and $S\subset A$ a ...
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### Question concerning commutative algebra

Let $R$ be a commutative ring. Let $P\subset R$ be a minimal prime ideal. Let $S=R-P$. Let $x\in P$ and $s \in S$. Is there any property saying that if $sx=0$, then $x=0$? Should anyone helps me, ...
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### $X$ compact Hausdorff implies $x\mapsto \mathfrak{m}_x$ is a homeomorphism

Let $X$ be a compact Hausdorff space. Denote by $\mathfrak m_x$ the prime ideal of $C(X)$ comprised of functions vanishing at $x$. Topologize $\operatorname{MaxSpec}C(X)$ with the initial topology ...
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### Divisors and prime elements in rings

Suppose $p$ is a prime element in a commutative ring. Does this imply its only divisors are $1,p$? By definition $a\mid p\implies \exists b:ab=p\implies p\mid a\text{ or }p\mid b$. If $p\mid a$ then ...
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### When is the localization of a commutative ring a finitely generated projective module?

Let $R$ be a commutative ring and $M$ an $R$-module. The tensor product $(-)\otimes M$ has a left adjoint $(-)\otimes M^\ast$ for $M^\ast =\mathsf{hom}(M,R)$ iff $M$ is finitely generated projective. ...
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### Spectrum of $\mathbb{Z}^\mathbb{N}$

Is anything known about the spectrum of $\mathbb{Z}^{\mathbb{N}}$? Notice that the fiber of $\mathrm{Spec}(\mathbb{Z}^{\mathbb{N}}) \to \mathrm{Spec}(\mathbb{Z})$ at a non-zero prime ideal $(p)$ is ...
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### Can anybody show how primary decomposition of ideals is useful. Any application?

So far I have been able to only see some standard counter examples or examples of Lasker-Noether decomposition theorem ( primary decomposition ). Why should someone care to learn it? ( I am being ...
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### subrings of a polynomial algebra

Let $k$ be an algebraically closed field and let $k[x]$ be the polynomial algebra in the variable $x$. It is known that any subring of $k[x]$ that contains $k$ is a finite module over some polynomial ...
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### finite extensions of discrete valuation fields: A method to find a basis

Suppose that $L|K$ is a finite extension of discrete valuation fields. Namely $w$ is a discrete valuation on $L$ extending a valuation $v$ on $K$. Now consider the respective rings of integers ...
### $M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?
Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...