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

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Commutative Rings Inside Commutative Rings with Field properties

As I was working through an algebra textbook, I noticed that a field $A$ is a commutative ring. But is it possible for $A \subset B \subset C$ where $A$, $C$ are fields and $B$ is not (and all of them ...
4
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2answers
878 views

Explicit example of Koszul complex

Let $R$ be a commutative ring and $x$ and $y$ two elements in $R$. I want to construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes $$C_2=0\to C_1=R\xrightarrow{\ ...
2
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2answers
689 views

Maximal ideals in multivariate polynomial rings

Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to ...
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1answer
243 views

Linear projective varieties

Let $Y\subset\mathbb{P}_{k}^{n}$ be a projective variety. We say that Y is a linear subvariety if $I(Y)$ can be generated by linear polynomials. Now how I should show that $Y\subset\mathbb{P}_{k}^{n}$ ...
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4answers
860 views

Noetherian module implies Noetherian ring?

I know that a finitely generated $R$-module $M$ over a Noetherian ring $R$ is Noetherian. I wonder about the converse. I believe it has to be false and I am looking for counterexamples. Also I ...
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2answers
102 views

Show there exists a finite flat morphism

If $f \in k[x_1,...,x_n]$ is irreducible then show there is a finite flat morphism $k[x_1,...,x_{n-1}] \to k[x_1,...,x_n]/(f)$ (i.e. $k[x_1,...,x_{n}]/(f)$ is finitely generated and flat as a module ...
4
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1answer
170 views

About prime ideals partial derivatives of polynomials

Given a polynomial $f(x_1,... ,x_n)\in \mathbb{C}[x_1, ... ,x_n]$, we can formulate its (formal) partial derivative with respect to each of the $x_i$, say $f_{i}$. If $f\in \mathfrak{p}$ and $f_{i}\in ...
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1answer
98 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: ...
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1answer
99 views

Ring containing a Dedekind ring

Suppose I have two domains, $A\subset B$, where $A$ is Dedekind and $\operatorname{Frac}(A)=\operatorname{Frac}(B)$. I also know that $B$ is both integrally closed and has height $1$. Is $B$ ...
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1answer
111 views

Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
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5answers
1k views

Finite quotient ring of $\mathbb Z[X]$

Since userxxxxx (I don't remember the numbers) deleted his own question which I find interesting, let me repost it: Let $f,g\in\mathbb Z[X]$ with $\mathrm{gcd}(f,g)=1$. Prove that the ring ...
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2answers
565 views

Which field is this quotient of a local ring by its maximal ideal?

Let $p\in\mathbb{Z}$ be a prime number, $\mathfrak{p}\subset \mathbb{Z}$ be the prime ideal it generates and let $\mathbb{Z}_{\mathfrak{p}}$ be the localization of $\mathbb{Z}$ at $\mathfrak{p}$, i.e. ...
3
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1answer
44 views

Null map between exact sequences

Is it true that if I have exact sequences of abelian groups $0 \rightarrow A\rightarrow B \rightarrow C\rightarrow 0$ and $ A_1\rightarrow B_1 \rightarrow C_1 $ (exact only in the middle) and ...
3
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3answers
336 views

Atiyah - Macdonald Exericse 9.7 via Localization

I am trying to show that the quotient of a Dedekind domain $A$ by an ideal $\mathfrak{a}$ is a PIR (principal ideal ring). Now by using the Chinese Remainder Theorem and the fact that a direct product ...
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1answer
814 views

Constructing Idempotent Generator of Idempotent Ideal

Exercise 2.1 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a commutative ring and $I$ an ideal that is finitely generated and $I=I^2$. Then $I$ is generated by an idempotent." ...
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1answer
185 views

Non-Free Finitely Generated Injective Modules over a Local Ring

I was wondering if someone could be so kind as to provide an example of a local ring $ (R,\frak{m}) $ and a non-free finitely generated injective module over $ R $. Thank you very much! I tried ...
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1answer
234 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
11
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0answers
242 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
6
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1answer
300 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
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1answer
1k views

UFD implies noetherian?

It is easy to show that a PID must be noetherian. My question is: Does UFD imply noetherian? If not, is there an easy counterexample? I apologize if this turns out to be a simple question. ...
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0answers
107 views

Kähler differentials, smooth algebras (one specific question)

Let $A$ be a commutative algebra (of essentially finite type, perhaps I have to add other reasonable assumptions) over a field, and I assume that $\Omega^1(A)$ is a projective module. Then the ...
4
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2answers
117 views

Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete?

Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}_X$-algebras cocomplete?
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2answers
125 views

How to show $\bigcap_{m \textrm{: maximal ideal}} A_m=A$?

$A$ is an integral domain. For every maximal ideal $m$ in $A$, consider $A_m$ as a subring of the quotient field $K$ of $A$. Show $\bigcap A_m=A$, where the intersection is taken over all maximal ...
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0answers
123 views

Geometric interpretations of several algebraic concepts

I would like to have some geometric intuition for - Noetherian rings/modules - Local rings - Projective modules - Injective modules As an illustration of what I am looking for, I was told once that ...
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1answer
533 views

Using Nakayama lemma to prove that surjective implies injective.

$A$ is a commutative ring, and $f:M\rightarrow M$ is an endomorphism of $A$-modules which is surjective. If I know that $M$ is finitely generated, I want to prove that $f$ is also injective. ...
6
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3answers
2k views

An ideal that is radical but not prime.

I'm preparing for an exam and, as part of this preparation, I'm looking for an ideal $I$ in an integral domain $R$ that is radical but not prime. Here is an example I'm fooling around with: ...
2
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0answers
73 views

Reduction of maximal graded ideal

Let $R$ be a commutative graded ring, $m$ be its graded maximal ideal, $M$ be a finitely generated graded module over $R$. A homogeneous ideal $I\subseteq m$ is a $M$-reduction of $m$ if ...
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1answer
592 views

Automorphisms of an affine line over finite fields

Let $k$ be an infinite field, and consider the affine line $\mathbb{A}_k^1$ over $k$. We know that every isomorphism $\varphi:\mathbb{A}_k^1\longrightarrow\mathbb{A}_k^1$ is of the form ...
6
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4answers
316 views

Is this a property of an integral domain that is not a field?

I am working on a specific problem and I've almost got it solved. To solve it, however, I need to prove one last claim (if it is even true): Consider an integral domain $R$ that is not a field. ...
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0answers
180 views

Divisorial ideal of a Krull domain

Now I try to do exercise 12.4 in the book "Commutative ring theory" by H. Matsumura. Let $A$ be a Krull domain, $I\subseteq \mathfrak p$ and $\mathfrak p$ is a height $1$ prime ideal of $A$. I don't ...
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1answer
119 views

Question about extensions of homomorphisms

I have difficulty understanding the proof of Theorem 3.2 in Lang's Algebra Chapter VII. Let $A$ be a subring of a field $K$ and let $x\in K, x\neq 0$. Let $\phi:A \rightarrow L$ be a ...
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1answer
99 views

Elementary questions about regular rings and Zariski tangent spaces

So I've got 3 rather related questions, which all seem to be true, except maybe the third. I'm asking because I remember thinking about this in the past and encountering a difficulty with all 3. ...
2
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0answers
91 views

ring of invariants for E6 singularity

As an exercise for myself, I've been trying for some time to calculate the coordinate ring of the $E6$ surface singularity. That is, I have a 2d representation of the binary tetrahedral group ...
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2answers
1k views

Localization at Maximal Ideals

Suppose $A$ is a commutative ring with $1\neq0$ satisfying the property that $A_\mathbf{m}$ has no nonzero nilpotent elements for any maximal ideal $\mathbf{m}$, where $$A_\mathbf{m}=S^{-1}A\quad ...
4
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1answer
274 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
6
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1answer
298 views

Questions about subalgebras of finitely generated $k$-algebras

Let $k$ be a field (if necessary assume $k$ to be algebraically closed). Let $A$ be a finitely generated $k$-algebra and let $B$ be a subalgebra of $A$. Remark that $B$ doesn't have to be noetherian, ...
4
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1answer
167 views

Exercise from Matsumura about DVRs

Another result I would really appreciate some help with: Suppose $R$ is a DVR and let $K$ be its field of fractions. Let $L$ be a finite extension of $L$. Prove that any valuation domain inside of ...
4
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1answer
365 views

Algebraic independence and dimension of a variety

A set of polynomials $\{f_1,\ldots,f_m\}$ in $k[x_1,\ldots,x_n]$ are algebraically independent over $k$ iff for all polynomials $p \in k[y_1,\ldots,y_m]$, $p(f_1,\ldots,f_m) = 0$ implies that $p = 0$. ...
1
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1answer
194 views

On Spec($K\otimes_k k^{p^{-\infty}}$)

Let $k$ be a field of characteristic $p > 0$. Let $K$ be an extension field of $k$. Let $k^{p^{-\infty}}$ be the perfect closure of $k$. Then Spec($K\otimes_k k^{p^{-\infty}}$) is a one element ...
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1answer
167 views

Example of not right exactness of local cohomology functor

Let $M$ be a module over a commutative ring $R$, $\mathfrak a$ is an ideal of $R$. Define $\Gamma_\mathfrak a(M)=\lbrace m\in M\mid\mathfrak a^tm=0 \text{ for some } t\in \mathbb{N}\rbrace$. Then ...
3
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2answers
171 views

$xy\in (x^2,y^2)$ if $R$ is a Dedekind domain

I would really like to see a simple proof for the following question, if possible. Let $R$ be a Dedekind domain. Then, $xy \in (x^2,y^2)R$ for any $x,y$ in $R$. Also, show that this fails in ...
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0answers
83 views

Question on dimension

In some text book of Commutative Algebra, the authors defined the height of an ideal $I$ of a commutative ring $R$ is the maximal of length of a prime ideal chains : $\mathfrak{p}_{0}\subset ...
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1answer
312 views

Finitely generated free modules of infinite rank.

We know that for general modules over a commutative ring with $1$, you can't always extract a basis from a generating set. This makes me think that maybe there should be free modules of infinite ...
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1answer
120 views

Homomorphism from module to localization

I have a question about this statement: Let $M$ be an $R$-module of finite length and let $\mbox{Ann}(M)\subset P_1,...,P_k$ be maximal ideals. If $n\in\mathbb{Z}_+$ is such that $P_1^n\cdots ...
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4answers
110 views

How can I show some rings are local.

I want to prove $k[x]/(x^2)$ is local. I know it by rather a direct way: $(a+bx)(a-bx)/a^2=1$. But for general case such as $k[x]/(x^n)$, how can I prove it? Also for 2 variables, for example ...
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2answers
237 views

Is the localization of such a Noetherian domain at every maximal ideal Artinian?

My question here is inspired by this question here. I am not asking how to prove the problem there, but if an alternative approach is possible. Suppose $A$ is a Noetherian integral domain in which ...
7
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2answers
284 views

$\operatorname{Spec}(\mathbb{C}\otimes_\mathbb{R}\mathbb{C})$ has two points

Why does $\operatorname{Spec}(\mathbb{C}\otimes_\mathbb{R}\mathbb{C})$ have two points? I know that ...
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2answers
226 views

Uniqueness of the decomposition of finitely generated modules over Dedekind domains

If $R$ is a Dedekind domain and $M$ a finitely generated $R$-module, then $M$ splits as a direct sum of a torsion and a projective $R$-module. Is such a splitting unique? And what if we ask about ...
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0answers
558 views

Structure theorem of finitely generated modules over a PID

I want to prove the structure theorem of finitely generated modules over a PID using the primary decomposition in a Noetherian $R$-module $M$. Applying the results on primary decomposition to the ...
5
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0answers
184 views

The saturation of a monomial ideal

Let $ d >0 $ be an integer, and let $ I \subset K[x_1,...,x_n] $ be the monomial ideal $$I = ( x_1^{a_1}x_2^{a_2}...x_n^{a_n} : \ \sum_{i=1}^n a_i =d; \ a_i < d\ \forall i).$$ (a) ...