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

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proving a property of CM-regularity

Let $k$ be a field, $S = k[x_1,\dots,x_n]$ the polynomial ring, $m = (x_1,\dots,x_n)$ and $I$ a homogeneous ideal contained in $m^2$. Define $R = S/I$. For $p \in \mathbb{N}$ we say that $R$ is ...
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0answers
16 views

Is the radical a differential ideal

I know that if we have a differential ideal, then the radical of the differential ideal is also a differential ideal. But someone know how to prove that, without using the fact that ...
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1answer
21 views

transitivity of integral extensions

Let $T{\geq}S{\geq}R$ be commutative rings. I'm trying to prove that if $T$ is integral over $S$ and $S$ is integral over $R$ then $T$ is integral over $R$. Let $t$ be in $T$ so there exist ...
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2answers
117 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
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23 views

Ring of Convergent Power Series in R and C is a Local Ring

Let $k=\mathbb{C}$ or $\mathbb{R}$, and let k{x} denote the ring of power series with appropriate coefficients that are convergent around 0. Check that k{x} is a local ring. I have a similar ...
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23 views

Ring of Formal Power Series Over a Field is a Local Ring

Let k be a field and let k[[x]] denote the ring of formal power series with coefficients in k. Verify that k[[x]] is a local ring. Could someone give an example of what k[[x]] looks like for a ...
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1answer
13 views

The module $M/M_n$ is of finite length

I have this doubt from a proposition in Dimension theory from Atiyah Macdonald. The proposition is as follows. Let $A$ be a noetherian local ring, $\mathcal{m}$ be its maximal ideal, $\mathcal{q}$ an ...
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1answer
45 views

Does localization functor have both sides adjoint functors?

Let $A$ be commutative ring, and $S$ a multiplicative set. The localization $S^{-1}$: $A$-module $\rightarrow$ $S^{-1}A$-module. Functor $F$: $S^{-1}A$-module $\rightarrow$ $A$-module, regard $A$ as ...
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1answer
25 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$

Let $n$ be a square-free integer such that $n\equiv 0,2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
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2answers
49 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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45 views

exact sequence proof [on hold]

Let $R$ be a commutative ring and $0\to L\to M\to N\to 0$ be a sequence of $R$ modules. Let $A$ be a multiplicativity closed subset of $R$ so that we can consider the corresponding localisation ...
3
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1answer
53 views

Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
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0answers
16 views

Examples of algebras having a module basis

I'm looking for examples of associative $R$-algebras, for which an $R$-module basis can be specified. Of course, if $K$ is a field, then any $K$-algebra admits such a basis, but this dis not what I'm ...
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90 views

Why is a principal prime ideal of $\mathrm{PID}[x]$ not maximal?

Let $R$ be a PID and let $f(x)\in R[x]$ be an irreducible primitive polynomial. I want to show that the prime ideal $(f)< R[x]$ is not maximal. It would be enough to find a prime $p\in R$ ...
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0answers
50 views

Characteristic of ring $R_1\otimes\ldots\otimes R_k$

Let $R_1,\ldots, R_k$ be unital rings and $\otimes=\otimes_\mathbb{Z}$ and $\mathrm{chr}$ the characteristic. How can one see that $$\mathrm{chr}(R_1\otimes\ldots\otimes ...
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0answers
55 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is 1. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that $$sup_{\{B\}}\; \text{pd}\; ...
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1answer
56 views

question about typical proof of Krull Intersection Theorem

In Atiyah-Macdonald, in the proof of Theorem 10.17 (Krull's intersection theorem), the authors go through a 4-line chain of arguments to show that that the kernel $E=\bigcap_{n=1}^{\infty}\mathfrak ...
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1answer
40 views

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 ...
3
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1answer
85 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
48 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. [closed]

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
59 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 ...
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1answer
63 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
52 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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1answer
160 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 ...
2
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1answer
66 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 ...
3
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1answer
75 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
3
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1answer
62 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 $\mathbb{C}$
3
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2answers
37 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
56 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
37 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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0answers
37 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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2answers
40 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 ...
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35 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 ...
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2answers
56 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
28 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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1answer
21 views

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
47 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
61 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
74 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
52 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 ...
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0answers
13 views

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
55 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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2answers
51 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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2answers
59 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
26 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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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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2answers
80 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
73 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 ...
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2answers
64 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 ...
5
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0answers
47 views

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. $f$ ...