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

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33 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 ...
0
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0answers
25 views

Improvement of Buchberger's Algorithm

1) Let $S(F)$ be the subset of $(k[x_1,\dots,x_n])^s$ consisting of all syzygies on the leading terms of $F=\{f_1,\dots,f_s\}$. Then every element of $S(F)$ can be written uniquely as a sum of ...
2
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1answer
29 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
46 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 ...
0
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0answers
40 views

Reference request: Cartier divisors versus invertible sheaves by Kleiman

Please delete this question if it is deemed inappropriate. Could someone link me to the paper "Cartier divisors versus invertible sheaves" by Kleiman please? My library doesn't provide access to it. ...
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0answers
17 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
70 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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0answers
56 views

Rank of finite ring extension, number of solutions of polynomial equations

While working though a graph theory paper, there was a construction heavily relying on ring theory and the authors mentioned the following "fact" without further introduction. Let $K$ be an ...
0
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1answer
27 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
61 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
69 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 ...
11
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0answers
106 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 ...
1
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1answer
51 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 ...
0
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1answer
64 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
192 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 ...
1
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1answer
19 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 @>>> ...
0
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0answers
50 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
41 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: ...
3
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4answers
88 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?) ...
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0answers
71 views

Finding the primary decomposition of an ideal $I$ and the associated primes of $A/I$

I've the ring $R=\mathbb Z[2X,X^2,X^3$] and the ideal $I=(2X,X^2)$. I'm trying to find: the associated primes of $A/I$ and $A/I^2$, and the primary decompositions of $I$ and $I^2$. How should I ...
2
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1answer
69 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
83 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
237 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
79 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
27 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 ...
2
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2answers
36 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 ...
4
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2answers
102 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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1answer
49 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 ...
4
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1answer
98 views

$M$ 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
38 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 ...
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2answers
101 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 ...
2
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1answer
23 views

Why does passing to the reduced ring not change the number of primes ideals?

I'm reading a note of Hochster's, and I don't follow something. He writes as the Corollary on page 9, Let $K\subseteq S$, where $K$ if a field, and $S$ is a finitely-dimensional $K$-vector space ...
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0answers
21 views

Question on Lemma preceding Going Up Theorem.

I have a question about Proposition 2.2.1 here: http://www.math.uiuc.edu/~r-ash/ComAlg/ComAlg2.pdf The proof has $S/R$ an integral extension of rings, $P_1,P_2$ prime ideals in $S$ lying over a prime ...
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0answers
68 views

$R_S (=K \cap A_{K,S})$ is a Dedekind domain

Let $K$ be a global field and let $S$ be a finite, nonempty set of places of $K$ containing the infinite one. Show that $R_S (=K \cap A_{K,S})$, the ring of $S$-integers of $K$, is a Dedekind domain. ...
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3answers
61 views

Idempotents in $\mathbf{CRing}$

I'm not able to find an example of an idempotent morphism different from an identity in the category of commutative rings with unity (an idempotent, as a morphism in that category, must preserve 1, ...
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1answer
66 views

Integral over a union of maximal two ideals

Let $A$ be Dedekind domain and $m_1$ and $m_2$ be maximal ideals of $A$ such that $A/m_1 \cong A/m_2$. How can I find a $x \in A-\{m_1 \cup m_2\}$ such that $x$ is not a root of any monic polynomial ...
2
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0answers
43 views

Eisenbud Corollary 6.7

Let $k$ be a field, $R=k[t]$ the polynomial ring in one variable, let $S$ be a Noetherian ring flat over $R$, If the fiber $S/tS$ over $t$ is a domain, and $U$ the set of elements of the form $1-ts$ ...
1
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1answer
31 views

on the statement of Theorem 3.3.7 in Bruns&Herzog

Let $\phi :(R,m) \rightarrow (S,n)$ be a local homomorphism of local Cohen-Macaulay rings, where $S$ is a finite $R$-module. In their proof of Theorem 3.3.7, Bruns&Herzog write that $\dim S = ...
9
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1answer
175 views

$\Bbb{R}/n\Bbb{Z}$ is isomorphic to $A_\Bbb{Q}/(\Bbb{Q}+C_n)$.

Let $A_\Bbb{Q}$ be the adele group of $\Bbb{Q}$. Let $C_n=\{x \in A_\Bbb{Q}: x_\infty=0 \text{ and }x_p \in p^{\operatorname{ord}_p(n)}\Bbb{Z}_p \text{ for prime }p\}$. I want to show that ...
3
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1answer
49 views

Tensor product and localisation

Let $k$ be an algebraically closed field and $K$ an extension field of $k$. Suppose $A$ is a finitely generated $k$-algebra which is a domain. Then we have a natural map $A \rightarrow A \otimes _ k ...
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1answer
21 views

showing that the Krull dimension of an extension module is zero

Let $(R,m,k)$ be a Cohen-Macaulay ring of dimension $d>0$ and let $M,C$ be CM $R$-modules such that $\dim M = 0, \dim C = d$. In the proof of Proposition 3.3.3-b(ii) in Bruns & Herzog, the ...
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1answer
54 views

Example of a module $M$ such that $\operatorname{depth}_{\mathfrak p}M<\operatorname{depth}_{A_{\mathfrak p}}M_{\mathfrak p}$; Matsumura, Ex. 16.5

I am looking for an example of a module $M$, a ring $A$, and a prime ideal $\mathfrak p$ such that $\operatorname{depth}_{\mathfrak p} M < \operatorname{depth}_{A_{\mathfrak p}} M_{\mathfrak ...
1
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1answer
55 views

Proof of the Auslander-Buchsbaum formula in Matsumura

There is a proof of Auslander-Buchsbaum formula in Matsumura's Commutative Ring Theory page 155. I am trying to understand the case $\operatorname{pd} M = 1$. He says take a short exact sequence $$ 0 ...
1
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1answer
70 views

Monic polynomials and integral elements.

Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Prove that the coefficients of $f$ and $g$ are integral over $A$. My attempt was to prove ...
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1answer
21 views

Converse of the implication $V(S)\subseteq V(T)\iff T\subseteq\sqrt{\langle S\rangle}$.

I'm having trouble recalling one direction of the following bi-implication. Suppose $S,T$ are subsets of the polynomial ring $k[X_1,\dots,X_n]$ over an algebraically closed field. We have ...
4
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1answer
72 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching ...
1
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1answer
43 views

Extending rings

This is a problem I've made up, which I cannot unfortunately solve. Any help will be appreciated. Let $R$ be a commutative ring with unity and $\operatorname{char} R=0$. I want to find the ring ...
1
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0answers
37 views

Elimination theory in Hartshorne

Does anyone know a good reference for elimination theory (Theorem 5.7A) mentioned in Hartshorne? The reference he gives is Van der Waerden modern algebra volume two, but it didn't feel locally ...
0
votes
2answers
101 views

If a module is nonzero, then a localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
0
votes
2answers
81 views

Non-zero prime ideals are maximal in the ring of algebraic integers

Let $A= \{y \in \mathbb{C} :$ $y$ integral over $\mathbb{Z}$ }. Let $P\not=\{0 \}$ be a prime ideal of $A$. I am supposed to prove that $P$ is also a maximal ideal. But I cant make it, is this really ...