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

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51 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 ...
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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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60 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 ...
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0answers
67 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 ...
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105 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
46 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 ...
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1answer
61 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 ...
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1answer
186 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 ...
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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 @>>> ...
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0answers
48 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
35 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
87 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
68 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
67 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
82 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 $\Bbb{Z}[X_1,...,X_n]$.
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1answer
206 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
78 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
33 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 ...
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2answers
98 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
46 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 ...
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1answer
89 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 ...
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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
98 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 ...
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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
20 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
61 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 ones. Show that $R_S (=K \cap A_{K,S})$, the ring of $ S-$ integers of $K$, is a Dedekind ...
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3answers
60 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
64 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 $1-ts$ ...
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1answer
30 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 = ...
8
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1answer
137 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
48 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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0answers
19 views

Integral elements and monic polynomials [duplicate]

Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Is it true that the coefficients of $g$ and $f$ are integral over $A$? Please help me with ...
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1answer
52 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 ...
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1answer
52 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 ...
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1answer
69 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 ...
3
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1answer
60 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 ...
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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 ...
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0answers
35 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 ...
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2answers
100 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, ...
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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 ...
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2answers
46 views

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$?

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$? Background: I was trying to prove that if $R{^m}$ is ...
0
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1answer
55 views

Contraction of non-zero prime ideals 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 $A$. Prove that $P \cap \mathbb{Z} \not= \{0 \}$. Iam totally stuck here, it is given that $P$ ...
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1answer
30 views

A property about quasi-primary modules

It is a fact that any discrete valuation domain $R$ has the property "P" that any proper submodule $N$ of any $R$-module $M$ is quasi-primary, in the sense that $\operatorname{rad}(N:M)$ is a prime ...
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0answers
46 views

Fibres of an ideal sheaf , total spaces and torsion groups

My question concerns a common example, which seems to often appear as an example/counter-example. Let $k$ be a field and consider the ideal exact sequence of the structure sheaf $k(p)$ of a point $p$ ...
1
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1answer
24 views

A doubt on a proposition involving Goldman domains.

$(*)$ Let $S/R$ be an extension of domains. Assume that for some $a\in R$, the ring $R[a]$ is Goldman. Then I want to show that $a$ is algebraic over $R$, whence $R$ is also a Goldman domain. DEF A ...
3
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1answer
91 views

Is the ring of entire functions coherent?

Call a commutative ring $R$ coherent if for each $n\in \{1,2,3,\cdots\}$ and each $n$-tuple $(r_1, ..., r_n)$ in $R^n$, the kernel of the map $R^n\owns (s_1, \cdots, s_n) \mapsto r_1 s_1 +\cdots + ...