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

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Localization modules and primary decomposition

Let $(R,m)$ be a Noetherian local ring and $M$ be an $R$–module. Suppose that for all prime ideals $p$, $p$ not equal to $m$, the localization $Mp$ is the zero module. Show that there exists a $k$ ...
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1answer
39 views

Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...
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1answer
66 views

Llocal ring and zeervisors [on hold]

Let $(R,m)$ be a Noetherian local ring. Suppose that all $x$ belonging to $m-m^2$ are zerodivisors. Show that all element of $m$ are zerodivisors. My progress: My idea is to see that the set of ...
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2answers
17 views

How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

Let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know how to show that $(x,y-z)/(g)$ is a prime ideal in $R/(g)$, and whether it is maximal or not. Thanks for the help!
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36 views

Localization modules and primary decomposition [on hold]

Let $(R,m)$ be a Noetherian local ring and $M$ be an $R$–module. Suppose that for all prime ideals $p$, $p$ not equal to $m$, the localization $Mp$ is the zero module. Show that there exists a $k$ ...
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2answers
38 views

Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminate and $k[t]\subseteq k[x]_{x}$ is a $k$-algebra extension so that $k[x]_{x}$ is a finitely generated ...
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1answer
29 views

Is the sum of saturated ideals saturated?

In a graded ring $S=\oplus_{k=0}^{\infty}S_k$, denote $m=\oplus_{k=1}^{\infty}S_k$, call an ideal $I$ to be saturated if $I=\cup_{n=1}^{\infty}(I\colon m^n)$. Is the sum of two saturated ideals still ...
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1answer
40 views

Torsion free module over a PID is flat

Suppose a ring of integers $S$ is an extension of a ring of integers $R$ with $\mathfrak{q}$ a prime ideal in $S$ and $\mathfrak{p}=\mathfrak{q}^c$ in $R$. Is there a straightforward way of showing ...
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0answers
19 views

Directs sum in exact sequences

I have a question about my homework question: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N\to 0$ be a splitting s.e.s. where $\alpha$ has a retraction $s$ and $\beta$ has a section $r$. ...
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1answer
29 views

Is an extension of a discrete absolute value discrete too?

Suppose $L/K$ is a finite extension of fields, suppose $v$ is a non-archimedean absolute value on $L$ such that the restriction of $v$ on $K$ is non-trivial and discrete. Can we say that $v$ is ...
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1answer
44 views

The ideal for image of Segre embedding

How to show the ideal $(X_{ij}X_{kl}-X_{il}X_{kj})_{0\le i,k\le m, 0\le j,l\le n}\subset k[X_{ij}]_{0\le i\le m, 0\le j\le n}$ is radical? I can show the zero locus defined by the ideal is the image ...
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0answers
39 views

Generalization of Bezout Theorem to many-hypersurface case in Hartshorne's setting

I try to follow the ideas in Hartshorne's Chapter 1, Section 7. Suppose we have algebraic sets $Y_1,...,Y_l$, I try to define their intersection number $I(Y_1,...,Y_l)$ to be the leading term of the ...
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1answer
17 views

Localization of modules and minimal generating sets.

Let $A$ be a ring and $M$ a finite $A$-module; for $p \in \text{Spec} \space A$, write $\mathcal{K}(\mathfrak{p})$ for the residue field of $A_\mathfrak{p}$, and let $\mu (\mathfrak{p}, M)$ denote ...
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1answer
44 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in the ring $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
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1answer
52 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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18 views

General differentials operators (Grothendieck definition) and polynomial rings

Let $A$ be an algebra over some field $\mathbb{k}$. A linear map $f:A\to A$ is said to be a differential operator of an order $\le n$ if for all $a_0,a_1,\ldots a_n\in A$ we have ...
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0answers
52 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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4answers
108 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
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1answer
37 views

Showing local ring isomorphisms

This is a problem in K. Hulek's Elementary Algebraic Geometry. I figured out that $k[X]$ is the collection of polynomials of the form $f(x) + g(y)$ and also the local ring of an affine line at the ...
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0answers
17 views

GCD-Domain and proprieties

Let $A$ be a commutative GCD-domain (not necessary UFD or Bezout) and $a,b,c$ elements of $A$ such that $\gcd(a,b) = \gcd(b,c) = \gcd(a,c) = 1$. Is it true that $\gcd(ab,c) = 1$ ?
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1answer
33 views

An example of Noether normalization

Let $A=k[x_1,x_2]/(x_2^2-x_1^3+x_1)$. As an example of Noether normalization, determine elements $y_1,\ldots,y_m\in A$, algebraically independent over $k$, such that $A$ is a finite ...
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2answers
69 views

$k[x]/(x^n)$ module with finite free resolution is free

How to show a $k[x]/(x^n)$ module with finite free resolution is free? Suppose we have a exact sequence $k[x]/(x^n)^{\oplus n_1}\to k[x]/(x^n)^{\oplus n_{0}}\to M\to 0$, how do we get ...
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1answer
32 views

Surjective homomorphism of rings. Every ideal of B is an extended ideal of an ideal of A. [on hold]

Let $f: A \rightarrow B$ be a surjective homomorphism of rings. I have to prove that every ideal of $B$ is an extended ideal of an ideal of $A$. Thanks! :)
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2answers
57 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
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1answer
49 views

Show structure of a commutative ring in a tensor product

I need some help with this: Let $R$ be a commutative ring and $S$ and $T$ be commutative $R$-algebras. Show that $$ S \otimes T $$ has the structure of a commutative ring with multiplication: $$ (s ...
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2answers
31 views

Units and nilpotents in quotient ring. [on hold]

$A$ is a commutative ring and $N(A)$ is the nilradical of $A$. If $A/N(A)$ is a field, show that every $a \in A$ is invertible or nilpotent.
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Krull dimension of a finitely generated integral domain over $k$ is equal to the transcendence degree.

This theorem is from Matsumura (p.34) Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k ...
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1answer
13 views

About freeness of modules over the coordinate ring of an affine variety

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...
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0answers
27 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
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1answer
18 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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1answer
28 views

If $0\to M'\to M\to M''\to 0$ is exact, why does $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup \operatorname{Ass}(M'')$.

I'm stuck on a proof I'm reading. Let $0\to M'\stackrel{\mu}\to M\stackrel{\sigma}\to M''\to 0$ be a sequence of $A$-modules. Then $\operatorname{Ass}(M)\subseteq \operatorname{Ass}(M')\cup ...
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1answer
24 views

Factorization in Dedekind domains

Let $R$ be a commutative, Dedekind (and therefore Noetherian) ring with $1$. Let $I$ be a non-prime ideal of $R$, and let $a,b$ be elements of $R$ such that $a\not\in I,b\not\in I$ but $ab\in I$. Let ...
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2answers
74 views

Hartshorne II Prop 6.8

My weaknesses with commutative algebra are really slowing down my progress through Hartshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...
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23 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
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2answers
55 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 - ...
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2answers
50 views

Show that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian

Prove that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian. I know that to be Noetherian, every ideal is generated by finitely many elements or equivalently R ...
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0answers
46 views

UFD and relatively prime elements

I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry": Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist ...
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1answer
63 views

Commutative Algebra and Game Theory

Is there any relationship between commutative algebra and game theory? For example, have any tools in commutative algebra been applied to game theory? A text or reference would be ideal, but I'd be ...
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51 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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1answer
87 views

Matsumura Example 2 (16.E)

I am reading example 2 (16.E) of Matsumura's Commutative Algebra where he gives an example of a non-CM ring. Let $A = k[x,y]$ and $B = k[x^2, xy,y^2, x^3, x^2y, xy^2,y^3]$. Then $A,B$ have the same ...
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1answer
56 views

Isomorphism of ring localized twice - Atiyah Macdonald Exercise 3.3

I studied AM before studying universal properties. When I solved the following exercise, I had a tedious solution that involved dealing with elements. Let $ A $ be a ring with multiplicatively ...
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2answers
41 views

The krull dimension of $\Bbb{Z}$ and artinian rings

On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$. However, for artinian ...
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1answer
47 views

Do we have $\dim A = \max_{\mathfrak{p} : \operatorname{ht}(\mathfrak{p}) = 0} \dim A/\mathfrak{p}$?

Let $A$ be a ring (assume Noetherian if necessary). Then it is clear to me that we have $$ \max_{\mathfrak{p} : \operatorname{ht}(\mathfrak{p}) = 0} \dim A/\mathfrak{p}\leq \dim A.$$ However, I ...
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1answer
29 views

Associated primes of the completion of a ring

I am working through a proof somewhere, and I want to use this: Let $(R,m)$ be a local ring (Noetherian commutative) and let $M$ be an $R$-module. If $p$ is an associated prime of $M$, then there ...
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61 views

When is $A = k[x_1,\ldots, x_n]/I$ integrally closed?

Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal). Can someone suggest strategies for showing that a ...
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1answer
51 views

Why does Proposition 1.8 in Atiyah-Macdonald imply that the smallest prime $\mathfrak{p}$ containing a primary ideal is equal to its radical?

Proposition 4.1 in Atiyah-Macdonal states that the radical of a primary ideal is the smallest prime ideal containing the primary ideal. They start the proof claiming that showing the radical is a ...
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3answers
111 views

Polynomials over $\mathbb{F}_2$ with certain values in $\mathbb{F}_4$

Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u ...
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1answer
31 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
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1answer
29 views

What is wrong with my proof of a step in Artin's construction of algebraic closure?

I'm working through Atiyah & MacDonald, and there's an exercise basically asking you to fill in a certain step in Artin's construction of an algebraic closure for a given field. The question is ...
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0answers
88 views

Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...