# Tagged Questions

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

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### Zero divisors, nilpotents and units in the ring of functions $\mathbb{R} \to \mathbb{R}$

Let $R$ be the set of all real valued functions defined for all real numbers under function addition and multiplication. i have to show that all the zero divisors of $R$ all nilpotent elements of ...
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### Non-zero divisor in an integral domain

Let $R$ be an integral domain and $P$ a prime ideal. Let $x$ be an element such that $xP^{m-1}=P^m$ for some $m>0$. Is $P$ generated by $x$?
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### Sum of localization maps

In Eisenbud's Commutative algebra with a view..., he shows that if an $A$-module $M$ has a finite length, then the sum of localization maps at maximal ideals is an isomorphism: ...
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### Finding the Hilbert Function for a certain ring

Right now I'm trying to find the Hilbert Function , and the corresponding Hilbert Polynomial for the ring $M=k[x,y,z,w]/(x,y) \cap (z,w)$. I just finished reading the first chapter of Eisenbud, so I ...
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### When is the ring of continuous functions absolutely flat?

This question was created in a discussion. Let $X$ be a topological space. Denote by $C(X; \mathbb{R})$ the ring of real-valued continuous functions defined on $X.$ Characterize those compact ...
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### The number of prime ideals lying over a given prime ideal

Put $A = k[x]$, where $k$ is an algebraically closed field and $x$ is an indeterminate. Let $B$ be a ring and $f: A \rightarrow B$ be finite integral morphism. How can one show that the number of ...
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### Atiyah-Macdonald Ex8.6

Is there anybody can give a proof? I can prove "finite" only, but I cannot prove "bounded". Here is the exercise: Let A be a Noetherian ring and Q a P-primary ideal in A. Consider chains of primary ...
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### translation from French

A passage from Bourbaki's Algebre X reads, "... l'homothetie de rapport $a_1$ dans $\oplus_{i\geq0}I^iM/I^{i+1}M$ est injective,..." Here $M$ is an $A$-module and $I=(a_1,\ldots,a_n)\subset A$. ...
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### For a ring of char $p$ where $p>0$ is a prime, what does $R^{1/p}$ mean?

If $R$ is a ring of characteristic $p\gt 0$, what does $R^{1/p}$ mean? I am not sure how to search for it, since I don't know a name for it. From the notation, it seems to be a ring consisting of the ...
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### Proof that $\mathbb{Z}$ has no zero divisors

Everyone knows the rules of zero divisors like $$\forall \alpha,\beta\in\mathbb{R}\;:\;\alpha\cdot\beta = 0\Rightarrow\alpha=0\vee \beta=0.$$ But how can I prove it for $\mathbb{Z}$? My first try was ...
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### Local Coordinate Systems under Integral Extension

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be an integral extension of regular local rings of dimension $d$ (of course, $\varphi$ is a local homomorphism). Furthermore, assume that $A$ contains ...
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### A finite commutative ring with the property that every element can be written as product of two elements is unital

I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a ...
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### $M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$

Let $S.$ be a graded ring, finitely generated by degree 1 elements as a $S_0$-algebra. Let $M.$ be a finitely generated graded $S.$-module. There exists a natrual map ...
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### When is an extension of a prime also a prime?

Suppose $R$ and $S$ are domains, $S$ is integral over $R$, $R$ is integrally closed, $p_1$ and $p_2$ are primes in the domain $R$, $p_1$ contains $p_2$, $q_1$ is a prime in the domain $S$ and lying ...
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### Having trouble with just one line in a proof on why nonzero prime ideals are maximal in a Dedekind domain

http://planetmath.org/?op=getobj&from=objects&name=ProofThatADomainIsDedekindIfItsIdealsAreInvertible In the PlanetMath article above, in the second paragraph of the proof of the first lemma, ...
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### A condition for a subgroup of a finitely generated free abelian group to have finite index

Let A be a free Abelian group of finite rank and B be a subgroup of A such that $A=B+pA$ for some prime number p, then how to prove $B$ is a subgroup of finite index in A? And if $A=B+pA$ holds for ...
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### Did Zariski really define the Zariski topology on the prime spectrum of a ring?

The question is not: “Did Zariski really define the Zariski topology?” It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?” Here is the motivation. --- On page ...
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### Flat algebras and tensor product

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules. Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$? Thanks
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### Is the support of an Artinian module finite?

If $R$ is an Artinian ring then it has finite maximal ideals. If $M$ is an $R$-module Artinian. ($R$ be a commutative Noetherian ring). Then, is $Supp(M)$ finite? Thanks.
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### Does the minimal injective resolution have the smallest length?

Let $A$ be a Noetherian (not necessarily local) ring and $M$ a finitely generated $A$-moduel. Is the length of the minimal injective resolution of $M$ always equal to the injective dimension of $M$? ...
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### isomorphism between specific generated field and specific quotient ring — gap in a proof

$K'$ is a field extension of $F$, $h\in F[x]$, $h$ is minimal for $u'\in K'$, $F(u')$ is a field generated by $F\cup \{u'\}$, $K'=F(u')$. In [1. XIII. Galois theory. 2. ...
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### One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I'm beginning to learn some grothendieck's algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one ...
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### Contraction of maximal ideals in polynomial rings over PIDs

Let $R$ be a principal ideal domain which is not a field, and let $M$ be a maximal ideal of the polynomial ring $R[X_1,\dots,X_n]$. If $n=1$ it is very easy to see that $M \cap R \neq 0$. Is this also ...
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### Why is the Hessian of an irreducible polynomial not zero?

Let $k$ be an algebraically closed field, $\operatorname{char}k=0$, $F$ be an irreducible homogeneous polynomial of degree$>1$ in $k[X,Y,Z]$, and ...
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### The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...
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### Are the determinantal ideals prime?

I want to prove the determinantal ideals over a field are prime ideals. To be concrete: For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be ...
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### Proof that a certain derivation is well defined

I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf Suppose, $R$ is ...
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### Every set of $n$ generators of $A^{n}$ is actually a basis

Let $A$ be a commutative ring with $1$. It is a standard result that every set of $n$ generators of the free $A$-module $A^{n}$ is actually a basis. The proof uses tensor products. I was reading a ...
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### Regularity ascends from a Noetherian ring to a polynomial or power series ring over it

I am looking for a proof of the following statement: A Noetherian ring $R$ is regular if and only if $R[x]$ is regular if and only if $R[[x]]$ is regular. I am trying to understand the properties ...
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### Integral extensions: one prime lying over implies equal localization

Here's a problem from Matsumura's book "Commutative ring theory" page $69$. Let $A$ be a ring and let $A \subset B$ be an integral extension, and $\mathfrak{p}$ a prime ideal of $A$. Suppose that $B$ ...
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### Grothendieck spectral sequence

given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed ...
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### Dimensions of modules of the maximal compact subrings of locally compact fields

I have checked the list of similar titles, proposed by the site. I hope this is not a repetition. This question arises from a proof of a proposition in the book Basic Number Theory, as follows. ...
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### Algebra over a ring

Could someone point me to a proof which shows that an algebra over a ring can be presented as a quotient of a polynomial ring (in possibly infinitely many variables).
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### Localization of prime ideals

Let $A$ be a commutative ring with $1$. Suppose that $P \subseteq Q$ are prime ideals in $A$ and that $M$ is an $A$-module. Prove that the localization of the $A$-module $M_{Q}$ at $P$ is the ...
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### Injective modules and ring homomorphisms

If there is a ring homomorphism $A\rightarrow B$ and if $Q$ is an injective $A$-module, is it true that $Q\otimes_A B$ is an injective $B$-module? I don't think it's true but can't think of a ...
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### Smooth ring maps and the module of differentials

Suppose $A$ and $B$ are commutative Noetherian rings, and $A \to B$ is a finite type smooth map. Then it is well know that the module of Kähler differentials, $\Omega^1_{B/A}$ is a projective module ...