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

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1answer
95 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 ...
0
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0answers
37 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?
2
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1answer
54 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
37 views

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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1answer
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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
49 views

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 ...
3
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1answer
45 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
37 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 ...
4
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1answer
64 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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0answers
238 views

Rational quartic curve in $\mathbb P^3$

By using similar arguments to the ones from my answer to this question, I can prove that the homogeneous coordinate ring of the rational quartic curve in $\mathbb P^3$, that is, $$R = K[x_1, x_2, x_3, ...
2
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2answers
46 views

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 ...
3
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0answers
61 views

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
3
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1answer
57 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 $\mathbb{C}$
6
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3answers
362 views

Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
3
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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: ...
0
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2answers
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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2answers
41 views

Quotient ring is cyclic group implies every ideal is generated by 2 elements

I'm trying to solve the following exercise: Let $R$ be a commutative ring with identity. If for every ideal $\mathfrak{a} \neq 0$ of $R$ we have ($R/\mathfrak{a}$,+) is a cyclic group then ...
0
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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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2answers
39 views

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 ...
13
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1answer
338 views

Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
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2answers
65 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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0answers
22 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 ...
6
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2answers
71 views

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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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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0answers
30 views

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 ...
1
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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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2answers
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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1answer
34 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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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 $A_0$?
2
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1answer
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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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2answers
49 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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1answer
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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0answers
49 views

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 ...
1
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1answer
57 views

Reference-request for $Monomial\ Ideals$

I newly started to study the book Monomial Ideals by Jürgen Herzog, Takayuki Hibi, but it is difficult in some cases for a beginner like me. Is any other reference which have similar topics (part I) ...
2
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1answer
57 views

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 ...
0
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1answer
24 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 ...
4
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2answers
395 views

$R[[x]]$ for a Noetherian ring $R$?

Let $R$ be a Noetherian ring. How can one prove that the ring of the formal power series $R[[x]]$ is again a Noetherian ring? It is well-known that the ring of polynomials $R[x]$ is Noetherian. I ...
0
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2answers
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 ...
2
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2answers
54 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
2
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1answer
96 views

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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1answer
68 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) ...
3
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1answer
112 views

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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0answers
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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. $f$ ...
2
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1answer
47 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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0answers
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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 ...
0
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1answer
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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1answer
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 ?
2
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1answer
150 views

Intuition? how the author reaches the answer?

I've a question on 2 problems in this book: 2.4. Let $S = K[x_1, . . . , x_6]$. Let $f = x_1x_5 − x_2x_4$, $g = x_1x_6 − x_3x_4$ and $h = x_2x_6 − x_3x_5$. (a) Find a monomial order $<$ ...
3
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0answers
82 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
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0answers
76 views

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 ...