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

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The radical of differential ideal

Consider a field $K$ such as $\mathbb{Q}\subseteq K$ and the ring $K[x,y]$, with the derivation $\frac{d}{dx}$. The ideal $\mathfrak a=(x^2,y^2,2)$ is a differential ideal. I have a doubt trying to ...
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1answer
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Atiyah-Macdonald, Chapter 10, Proposition 10.15 clarifications

In Proposition 10.15 in Atiyah-Macdonlad, what does the equality $\hat{\mathfrak a}=\hat A\mathfrak a$ mean? I know that there is an isomorphism $\hat A\otimes_A\mathfrak a\cong\hat{\mathfrak a}$ and ...
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1answer
73 views

Algebraic vs. Integral Closure of a Ring?

Let $R\subseteq S$ be a ring extension. It is true that the set of elements of $S$ that are are integral over $R$ (i.e. satisfy a monic polynomial equation over $R$) is a subring of $S$. Can anyone ...
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3answers
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Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [on hold]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
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Ring of fractions $S^{-1}A$ and localisation

I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it. (a) Give an ...
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1answer
60 views

Need an explanation for homomorphism in commutative algebra

I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which ...
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1answer
43 views

Preparation for a graduate commutative algebra course based on Eisenbud

I am an undergraduate with two semesters of algebra(groups,rings, Galois theory, etc) under my belt and I am planning on going through Atiyah and MacDonald's book over the summer. Is this sufficient ...
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133 views

The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
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1answer
55 views

Why we can consider both modules as modules over $R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9)

I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem? ... i know that ...
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1answer
68 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
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1answer
60 views

Failure of Luroth's theorem for transcendence degree 3

Can somebody give an example which shows the failure of Luroth's theorem for transcendence degree 3 over $\mathbb{C}$
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2answers
36 views

What's the relation between prime spectrum and affine space?

Let $A$ be a ring ,$X$ be the set of all prime ideal of $A$.For each subset $E$ of $A$,let $V(E)$ denoted the set of all prime ideals of $A$ which contain $E$. we have: ...
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0answers
48 views

Regular sequence and projective module

Let $R$ be commutative ring and $x,y$ an $R$-regular sequence. Then I know that $ R/(x)$ is not a projective $R$-module. My question: Is $R^{2}/(x,y)R$ a projective $R$-module?
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1answer
36 views

Injective dimension and depth

Here is Bruns and Herzog's book Cohen-Macaulay Rings, Theorem 3.1.17: Let $R$ be a Noetherian local ring, and $M$ a finite $R$-module of finite injective dimension. Then $\operatorname{inj\ ...
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27 views

How do we show that the union of associated prime ideals of a reduced primary decomposition is equal to the set of zero divisors in R?

I am working on one of Commutative Algebra problem from Hungerford's Algebra (GTM 73). The problem is Exercise $VIII.4.8$ on page 394 and is: Let $R$ be a commutative Noetherian Ring with identity ...
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A proposition about valuation ring

Q1 $x \in m\Rightarrow x~\text{is a element of an ideal}\Rightarrow ax~\text{is a element of an ideal}\Rightarrow ax~\text{is a non-unit}$ What's the mean of $(ax)^{-1} \in B$ ? Q2 $x,y \in ...
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Is the relative ideal of two affine curves $C\subset Z$ a finite dimensional vector space?

Let $I$ be a (non necessarily radical) ideal in the ring $A=\mathbb C[x,y,z]$, with Hilbert function $h=T+n$, where $T$ is a variable and $n>0$ is an integer. Let us assume that $I$ is contained in ...
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2answers
55 views

valuation ring is a field?

suppose $a$ and $a'$ are units of $B$ ,$b$ and $b'$ are the elements of any ideal of $B$. $x$ is a element of $K$. $K$ consist of $a/a,a/b,b/a,b/b$ $\color{green} x=a/a' \Rightarrow x\in ...
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2answers
27 views

What the difference between $A/m$ and $A_0$

$A$ is a integral domain. $m$ is a maximal ideal . $A_0$ is the localization of $A$ by $A-0$.(Field of fractions) $A/m$ is the quotient at $m$. What the difference between $A/m$ and $A_0$?
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Counterexample to “A/I is Artinian, when I is the annihilator of Artinian A-module”.

Let M be an Artinian A-module and let I be the annihilator of M in A. Is A/I necessarily an Artinian ring? I believe the answer is no since this comes off of a similar result regarding Noetherian ...
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1answer
42 views

Can we replace the $B$ to $A$ in this proposition

I am working through Atiyah's Commutative algebra and am having question with the following proposition: $\text{Page 63:}$ Proposition 5.15. Let $A$ $\subseteq$ $B$ be integral domains, $A$ ...
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1answer
56 views

Surjective homomorphism on Commutative Ring

Let $A$ be a commutative ring, $R= A[x_{1},...,x_{n}]$ and $(a_{1},...a_{n}) \in A^{n}$ . Let $\phi : R \to A$ be defined by $\phi (f(x_{1},...,x_{n}) = f(a_{1},...,a_{n})$. Then show that $\phi$ is a ...
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Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,…,x_6]$ a radical ideal? Is it a prime ideal?

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,...,x_6]$ a radical ideal? Is it a prime ideal? thanks
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divisible modules

In surveying LMR of T.Y.Lam, I reached a paragraph stating that "when R is a domain every direct sum or direct product of divisible modules is divisible." My question is that "should R is not a ...
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showing that a set of linear forms is closed (Bruns and Herzog, Theorem 4.2.12)

Let $k$ be an infinite field and $R$ a homogeneous $k$-algebra, i.e. a $k$-algebra that is generated by linear forms. Let $s = \sup\left\{\dim_k h R_{n-1} : h \in R_1\right\}$, where $R_i$ denotes the ...
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2answers
50 views

Show module is Noetherian

Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre ...
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49 views

How is an onto map implies $N+mM=M$ in Commutative Algebra?

I am having hard time understanding some details in Proposition 2.8 which is on page 22 of Atiyah and Macdonald's book: Introduction to Commutative Algebra. How the writers are claiming that being ...
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49 views

Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are finitely many prime ideals $P$ ...
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1answer
25 views

Integral multiplicative system over a domain

Suppose $A$ is a domain and $S\subseteq A$ is a multiplicative system. Show that $S\subseteq A^\times$ if and only if $S^{-1}A$ is integral over $A$. I've started $\Leftarrow$ below... Suppose ...
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49 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
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66 views

Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...
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1answer
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Problem 10.5 in Atiyah's book

Here is the problem: Let $A$ be a Noetherian ring and $a$, $b$ be ideals in $A$. If $M$ is any $A$-module, let $M^a$, $M^b$ denote its $a$-adic and $b$-adic completions respectively. If $M$ is ...
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62 views

Examples of Cohen-Macaulay rings.

I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay: $k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$. Also I am looking for a ring which is ...
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Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds. $f$ ...
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1answer
69 views

Every radical is prime?

$a$ is an ideal of $A$. $$f:A\to A/a,\ \ x∈r(a)$$ r(a) is a prime ideal? proof 1: $x^n\in a$ for some $n \Rightarrow (x+a)^n\in a$ for some $n \Rightarrow f(r(a))=\text{nil-radical}$ in $f(a) ...
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1answer
48 views

Localization of a module as direct limit

Let $A$ be a commutative ring, $S \subset A $ a multiplicatively closed set and $M$ an $A$-module. For every $s \in S$ we denote by $M_{s}$ the localization of $M$ with respect to $\{ 1, s, s^2, ...
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Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
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is there a criterion that says whether an ideal is radical or not?

Let $R=k[x,y,z]$. Is there a criterion that says whether an ideal of $R$ is radical or not? thanks
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About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $

Let $\mathcal{N}{oetherian}\mathcal{C}\mathcal{R}{ng} \overset{def}{=} {\left\lbrace{ R \in \mathcal{C}\mathcal{R}{ng} \wedge R \,\text{is}\, \mathcal{N}{oetherian} }\right\rbrace}$. I define the ...
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Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
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97 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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What are the maximal ideals of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$?

I know that $\mathbb{Z}[t,t^{-1}]$ is a localization of $\mathbb{Z}[t]$, the multiplicative set consisting of the non-negative powers of $t$. But I do not know the maximal ideals of ...
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75 views

Is this ideal prime? [closed]

Let $A = k[X, Y, Z]/(XY - Z^2)$, where $k[X, Y, Z]$ is a polynomial ring over a field $k$. Let $x, y, z$ be the image of $X, Y, Z$ respectively by the canonical homomorphism $\phi\colon k[X, Y, Z] ...
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exercise of Matsumura about CM

I have 2 question about this exercise of Matsumura: question 1- why $y^3$ is $R/(x^3)$ regular? question 2- I hardly (in 20 lines) can prove is there a short way or intuition for this part ? ...
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42 views

exercise of Matsumura

my question is about this exercise of Matsumura: in the proof hint we use is this obvious? or e should define an isomorphism?
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60 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
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Is the length of the composition series of a free module identical to the number of its bases?

Let $A_0$ be an Artinian ring, $M$ a free $A_0$-module. Then, is the length of the composition series of $M$ identical to the number of its bases? It seems to me that it is not. If $\mathfrak a$ ...
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1answer
44 views

If $S$ is the integral closure of $R$ in it's field of fractions and $S\subset R_{m}$ is $R_{m}$ integrally closed?

Let $R$ be a domain and $K$ be it's field of fractions. Let $S$ be the integral closure of $R$ in $K$. Let $M$ be a maximal ideal of $R$. If $S\subset R_{M}$ is $R_{M}$ integrally closed in $K$? My ...
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57 views

Completion of integral domain

Let $A$ be an integral domain with the $I$-adic filtration. Let $B$ be the fraction field of $A$. My question is the following: Is the fraction field of the completion of $A$ the same as the ...
5
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1answer
60 views

Can a ring isomorphism change the structure of a module?

Let $M$ be an $R$-module, where $R$ is a ring with unit. Given a ring automorphism $\phi: R \rightarrow R$, we can define a new $R$-module structure on $M$ by $r \cdot x = \phi(r) x$ for all $r \in ...