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

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Inversion of an element in Picard group over commutative ring

I'm having some troubles understanding a proof in Commutative Algebra Chapter I - VII of N. Bourbaki. It's on pag 114 of the book. Here's what it says: Theorem 3 ... (ii) Conversely, if $M$ ...
1
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1answer
62 views

Maximal among some ideals is prime

I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy ...
8
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3answers
290 views

Are Dummit and Foote making a mistake in proving Cohen's theorem?

Exercise 11 on page 669 (this is Chapter 15) wants to prove Cohen's theorem that if every prime ideal of a ring is f.g. then every ideal is f.g. that is the ring is noetherian. The highbrow (perhaps?) ...
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0answers
37 views

$S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$

Behold any commutative ring $R$. Is it true that $S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$ for any multiplicative subset $R$ of $S$? I couldn't find this in full Bourbaki generality, not ...
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0answers
64 views

Calculating the Hilbert polynomial of a principal ideal

If we have a field $K$, and a homogeneous polynomial $f \in R=K[x_1, \ldots, x_n]$, then the ideal generated by $f$ is a graded module over $K$, and we can calculate its Hilbert polynomial. (I am ...
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1answer
165 views

Tensor product of the fraction field of a domain and a module over the domain

Given a fraction field $k(x)$ of the polynomial ring $k[x]$ over a field $k$ and an integral domain $R$ that is also a $k[x]$-module, is it true that $k(x) \otimes_{k[x]} R \cong Frac(R)$? I ...
2
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1answer
73 views

Cohen-Macaulay ring and module: R-regular vs M-regular

Let $R$ be a Cohen-Macaulay ring and $M$ be a finite generated maximal Cohen-Macaulay module. I know that the R-regular sequence must be $M$-regular. Here are my questions: 1) Must an $M$-regular ...
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1answer
53 views

On Prime and Maximal Ideals in a Commutative Ring with Unity

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal ...
5
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2answers
131 views

Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)?

Let $R$ be a local ring containing a field isomorphic to its residue field $k$. Assume $R$ is a localization of a finitely-generated $k$-algebra. Can $\Omega_{R/k}$ be free of rank $r\neq\dim{R}$? ...
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1answer
64 views

Dedekind domain necessary for equivalence of flatness and torsion-free

It is well-known that for finitely generated modules over a Dedekind domain, flatness and torsion-free are equivalent. Is this true for general Noetherian rings? If not, where is the dimension one ...
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2answers
96 views

Product of ideals for Nakayama's Lemma

The result to be proved is the following: Let $R$ be a local Noetherian ring. Then the minimum number of generators of the unique maximal ideal $P$ equals the dimension of $P/P^2$ as a vector space ...
2
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1answer
54 views

System of polynomial equations and Nullstellensatz

Let $k$ be an algebraically closed field and the field $K$ contains $k$. I am trying to prove that if $F_1,...,F_m\in k[x_1,...,x_n]$ and the system of polynomial equations $F_1=0,...,F_m=0$ has a ...
2
votes
1answer
276 views

Integral closure in field of fractions.

Let $I$ be the ideal generated by $2xy+x^2+y^3$ in $\mathbb{R}[x,y]$. Define $A:=\mathbb{R}[x,y]/I$, I want to find the normalisation of $A$, that is, the set $B= \{ a \in \text{Frac} A : \text{a ...
2
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2answers
538 views

Commutative ring is semisimple iff it's isomorphic to a finite direct product of fields.

I am trying to prove the following: Let $R$ be a commutative ring. Prove that $R$ is semisimple if and only if it is isomorphic to a direct product of a finite number of fields. Suppose $R$ is a ...
4
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0answers
72 views

a subtle detail in the proof of Theorem 3.3.7 of Bruns and Herzog

Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ ...
2
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1answer
62 views

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(\operatorname{rad}A)=\operatorname{rad}B$. I know that if $A$ is a semilocal ring and if $I_{1},\dots, ...
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0answers
56 views

Is there any relationship between localization and completion of a module?

Let $R$ be a commutative ring, $\mathfrak p$ a prime ideal of $R$ and $M$ an $R$-module. I've seen the terms 'localization' $M_\mathfrak p$ of $M$ and the completion $M_\mathfrak p$ at $\mathfrak p$ ...
3
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1answer
42 views

valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
4
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3answers
240 views

Recommendations for Commutative Algebra Software?

I'd like a software that I can use to work with commutative algebra, specifically to figure out S-Polynomials, Buchberger's Algorithm, etc. I have Mathematica; if anyone could refer me to a package, ...
2
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0answers
109 views

Noetherian normal ring is a finite direct product of normal domains

Let $A$ be a Noetherian normal ring, that is, the localization of $A$ at every prime is a normal domain. I want to show $A$ is a finite product of normal domains. If $p_1,\ldots,p_n$ are the ...
0
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1answer
156 views

Why does a ring homomorphism induce a continuous map between spectra? [duplicate]

Let $\varphi: A \rightarrow B$ be a ring homomorphism. Let $f =\mathrm{Spec}(\varphi) : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ be the map associated to $\varphi$. Why is the map $f$ is continuous? ...
2
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0answers
30 views

Improvement of Buchberger's Algorithm (second part)

Suppose $S_j$ is a homogeneous syzygy of multidegree $\gamma_j$ in $S(G)$, where $G=\{g_1,\dots,g_t\}$. Show that $S_j G=\Sigma_{i=1}^{t} c_ix^{\alpha(i)}g_i$ has multidegree $< \gamma_j$. Now, I ...
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1answer
83 views

Question concerning the chinese remainder theorem for commutative rings

let $S$ be a commutative ring and $I_1,...,I_n\unlhd S$, such that $I_i+I_j=S\ \forall i\neq j$. Let $g_1,...,g_n\in S$. Why are there $h_1,...,h_n,h'\in S$, such that ...
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0answers
44 views

Radical of an ideal in a finitely generated ring over $k$ is the intersection of maximal ideals containing it. [duplicate]

From Matsumura p.34 Let $k$ be a field, $A$ a ring which is finitely generated over $k$, and $I$ a proper ideal of $A$; then the radical of $I$ is the intersection of all maximal ideals containing ...
2
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1answer
31 views

2 questions concerning identities of closed subspaces of $spec(S)$ for a commutative ring $S$

I have the following questions: Let $S$ be a commutative ring and let $M,N$ be closed subspaces of $spec(S)$, such that $M\cap N=\emptyset$. 1) Why are there ideals $I_1,I_2\unlhd S$, such that ...
2
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1answer
56 views

integral ring homomorphism

Consider a homomorphism $f: A\to B$ of commutative rings and let $b\in B$. Let $g\colon A[X]\to B[X]$ be defined by $g(X) = X$. Put $I = g^{-1}((bX-1))$ (contraction of the ideal $(bX-1)\subseteq ...
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0answers
20 views

Inverse image of a maximal ideal under a morphism of finitely generated $\mathbb{C}$-algebras. [duplicate]

Let $$ f: A\to B $$ be a morphism of finitely generated $\mathbb{C}$-algebras, suppose $\mathfrak{m}\unlhd B$ is a maximal ideal, I want to show that $f^{-1}(\mathfrak{m})$ is a maximal ideal of $A$. ...
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1answer
78 views

Finding a coordinate ring

I am having hard time in calculating (or constructing) $\displaystyle\frac{\mathbb C[x,y]}{\langle y^2 - x^3 - x\rangle}$. I tried homogenizing the ideal $y^2 - x^3 -x $ to $ wy^2 - x^3 - xw^2$. But ...
0
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1answer
52 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
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0answers
102 views

Irreducibility of a polynomial and connectedness of its zero set

Let $P$ be a polynomial in $\mathbb{C}[z_1,z_2,...,z_n].$ Let $Z(P)$ denotes its zero set in $\mathbb{C}^n.$ I have the following question: Does the irreducibility of $P$ imply that $Z(P)$ is ...
0
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0answers
81 views

Induced Spec map for a morphism of finitely generated $\mathbb{C}$-algebras

I have a morphism $f:A\longrightarrow B$ of finitely generated $\mathbb{C}$-algebras. I have proven, using Zariski's lemma, that the inverse image of a maximal ideal $M \subset B$ is a maximal ideal ...
12
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0answers
151 views

How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
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1answer
89 views

Saturation of a power of an ideal

Let $k$ be a field and let $R=k[x,y,z]$ and $\mathfrak m=(x,y,z)$. Let $I$ be a graded ideal of $R$. For all $n\in \mathbb{N}$ on has $$ (I^{\rm sat} )^n\subset (I^n)^{\rm sat},$$ where $$I^{\rm ...
1
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1answer
76 views

Castelnuovo-Mumford regularity and postulation numbers

I have a problem about Castelnuovo-Mumford regularity. This is a proposition from Castelnuovo-Mumford regularity, relation types and postulation numbers by M. Brodmann and C. H. Linh. My ...
3
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1answer
347 views

Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor

I don't remember whether there was a special name for a commutative ring where every non-invertible element is a zero-divisor. And I also forgot the different ways in which a non-invertible element ...
2
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1answer
60 views

Boundary Homomorphism

I was studying the proposition 2.10 of Atiyah and MacDonald's Introduction to Commutative Algebra, and have a question. The proposition says: Let $$ \require{AMScd} \begin{CD} 0 @>>> ...
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0answers
65 views

Non finitely-generated projective $\mathbb{Z}$-module [duplicate]

Let $M$ be a projective $\mathbb{Z}$-module. Must $M$ be free? It is easy to see that the answer is yes if $M$ is finitely generated, but I do not know about the general case. If the answer ...
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1answer
50 views

How can associated primes be distributed among submodule and quotient?

Suppose $M$ is a finitely generated module over a Noetherian ring $A$ (commutative, with identity). Then for a submodule $N$, we have the following relation among the sets of associated primes: ...
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4answers
105 views

If every free $R$-module has the property that independence implies extendibility, is $R$ necessarily a field?

Definition. Whenever $M$ is a free $R$-module, let us call a subset $A$ of $M$ extendible iff there is a basis $B$ for $M$ such that $A \subseteq B$. (Is there a standard name for this condition?) ...
5
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1answer
112 views

Exact sequence of $A$-modules [duplicate]

I was trying to demonstrate the Proposition 2.9 of Atiyah and MacDonald's Introduction to Commutative Algebra. But I couldn't do the following: Let $M$, $M'$, and $M''$ be $A$-modules, $v$ and $u$ ...
0
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1answer
100 views

Property of prime ideals of $\Bbb{Z}[X_1,…,X_n]$

Let $P$ be a prime ideal of $\Bbb{Z}[X_1,...,X_n]$. How to show that there exist a prime number $p$ such that $(p)+P$ is not $\Bbb{Z}[X_1,...,X_n]$.
5
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1answer
261 views

The germ induced by an irreducible polynomial

Let $P\in\mathbb{C}[z_1,z_2,\ldots,z_n]$ be an irreducible polynomial. Let $a\in\mathbb{C}^n$ be such that $P(a)=0.$ Consider the germ of holomorphic functions at the point $a,$ denoted by ...
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0answers
107 views

Hartshorne, Exercise 3.18, Chapter 2

Let $B$ be a noetherian integral domain, let $A$ be a subring of $B$ such that $B$ is a finitely generated $A$ algebra. Assume that $A$ is also noetherian. Let $b$ be a non-zero element of $B$. How ...
1
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1answer
29 views

Fraction ring contains another implies prime contains another

As part of lemma 6.4 in Hartshorne, I came across a statement that I can't prove Let $m,n $ be maximal ideals of an integral domain $A$. Then $ A_m \subset A_n$ implies $n \subset m $. It is ...
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2answers
63 views

Fibers of $\operatorname{Spec}(R)\to\operatorname{Spec}(S):\mathfrak{q}\mapsto \mathfrak{q}\cap S$ are discrete?

Suppose $S$ is a subring of a commutative ring $R$, such that $R$ is finitely generated as an $S$-module. I"m curious about a property of the map ...
5
votes
2answers
438 views

Importance of Noether normalisation lemma

The Noether normalization lemma states that if $k$ is a field, and $A$ a finitely generated $k$-algebra, then there exist elements $z_1,...,z_m \in A$ such that (i) $z_1,...,z_m$ are algebraically ...
2
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2answers
156 views

Associates in the ring of continuous real-valued functions on $[0,1]$

I have tried to give a proof of the following theorem but I feel very unsure and would be very grateful if someone would check it for me Many thanks! Theorem. Let $R$ be the ring $C[0,1]$ of ...
5
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1answer
146 views

Generic freeness: $M$ f.g. projective, then there is $a\notin \mathfrak p$ for which $M[a^{-1}]$ is a free $R[a^{-1}]$-module.

In Jacobson's BAII, he aims to show that any finitely generated projective module over a connected ring has a rank, where he defines this as follows: First, he shows that any finitely generated ...
0
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1answer
60 views

Simple question on tensoring by a quotient ring

$A \subset B$ is an extension of commutative rings s.t. $B$ is a f.g. free $A$-module of rank $n$, so I have $A^n \stackrel{\sim}{\longrightarrow} B$ as $A$-modules. Let $\mathfrak a$ be an ideal of ...
10
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2answers
159 views

Contents of Tor modules

I'm interested in knowing a concrete description of what elements of Tor modules $\mathrm{Tor}^i_R(M,N)$ "are". As it stands I have no real intuition for, say, maps between Tor modules induced by ...