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

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4
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1answer
45 views

Is radical of finitely generated ideal finitely generated?

Let $R$ be non-noetherian commutative ring with identity and $I$ be a finitely generated ideal of $R$; say $I = (a_1, \cdots, a_n)$. Question.1 Is $\sqrt I$ necessarily finitely generated? ...
5
votes
1answer
74 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
-1
votes
1answer
70 views

If $A$ is a maximal ideal, then $\mathbb{F}_p[x,y]/A$ is a finite field [closed]

Let $A$ be a maximal ideal of $\mathbb{F}_p[x,y]$. Then $\mathbb{F}_p[x,y]/A$ is a finite field. PD: I cannot show that it is necessarily finite.
0
votes
2answers
70 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
1
vote
1answer
74 views

Prove that in the ring $F[t,t^{-1}]$ we have $x=t^n \Leftrightarrow x \mid 1$ and $t-1 \mid x-1$

I want to prove the following lemma: For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and ...
1
vote
1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...
2
votes
1answer
78 views

Is direct limit of local rings a local ring?

Let $\{R_i\}_{i\in A}$ be a directed set of commutative local rings with directed index set $A$, and let $R$ be the direct limit of this set. I want to know if $R$ is a local ring (we know that $R$ is ...
3
votes
2answers
54 views

Constructing DVR's from arbitrary UFD's

Is the following statement true? Let $A$ be an UFD and $p\in A$ prime, then $A_{(p)}$ is a discrete valuation ring. I think yes: For every element $x$ of $Q(A_{(p)})=Q(A)$, there is a unique ...
2
votes
1answer
38 views

Why do we need injectivity in the definition of integral dependence?

Let $f: A \rightarrow B$ a ring morphism of commutative rings, then one has on $B$ a multiplication by elements of $A$ defined by $b*a \doteq b.f(a)$ (where . is the multiplication in the ring $B$). ...
1
vote
0answers
88 views

On a theorem of Akizuki concerning the minimal number of generators of an ideal

I am looking for a theorem of Akizuki I was told by my professor. He said me that Akizuki showed in his paper "Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz" (1938) a result ...
3
votes
0answers
107 views

When will $A[x_1, \ldots, x_n]$ satisfy the dimension formula?

What property should $A$ satisfy so that $A[x_1, \ldots, x_n]$ satisfies the dimension formula, $$\mathrm{dim}(A[x_1, \ldots, x_n]) = \mathrm{dim}(A[x_1, \ldots, x_n]/\mathfrak{p}) + ...
1
vote
1answer
52 views

Is the ring $A[x_1, \ldots, x_n]$ Cohen-Macaulay? Does the dimension formula hold?

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over a Noetherian, commutative ring, $A$. Is the polynomial ring Cohen-Macaulay? If not, does it follow the dimension formula, $ \mathrm{dim} (A[x_1, ...
4
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0answers
71 views

What properties $R \subseteq S$ should have in order that every prime ideal of $S$ is extended?

My question is almost the same as In what conditions every ideal is an extension ideal?; I allow myself to ask this question, since there is no answer to the above question. My question: Given ...
2
votes
2answers
48 views

If $A\otimes_k l$ is a normal integral domain then $K(A)\otimes_k l$ is a field.

I am trying to solve Ex. 5.4.M in Vakil's notes. Quoting the text: Suppose $A$ is a $k$-algebra, and $l/k$ is a finite extension of fields. (Most likely your proof will not use finiteness; this ...
3
votes
1answer
60 views

Stalks of the sheaf of total quotient rings

Let $X$ be a scheme, for each $U$ open in $X$, let $S(U)$ be the set consisting of elements of $O_X(U)$ whose image in $O_{X,p}$ is a non-zerodivisor for every $p\in U$. In particular, if $U = ...
3
votes
1answer
90 views

If $A$ is a finitely generated $R$-module, is $\operatorname{Hom}_R(A,R)$ finitely generated? [duplicate]

Let $R$ be an utterly arbitrary commutative, unital ring. Let $A$ be a finitely generated $R$-module. Is $\operatorname{Hom}_R(A,R)$ finitely generated as an $R$-module? Intuitively and based on ...
0
votes
1answer
17 views

Height of associated prime ideal is zero

Let $P\in\operatorname{Ass}(0)$ in a Noetherian ring $R$, and assume the local ring $R_P$ is a domain. I want to prove that the height of $P$ is zero. I know that in a Noetherian ring, each ideal ...
2
votes
3answers
77 views

Non-domain of Krull dimension zero

Let $F$ be a field, and $V$ be an $F$-vector space. Make $R=F⊕V$ a ring by putting $xy=0$ for $x,y\in V$. Is it true that the Krull dimension of $R$ is equal to zero? If this is so, $R$ would be an ...
1
vote
2answers
53 views

Systems of Parameters are exactly $R$-sequences

If $(R,m)$ is a local Cohen-Macaulay ring, it is well-known that each system of parameters is an $R$-sequence. Is any $R$-sequence (in a Cohen-Macaulay ring) a system of parameters? I am aware ...
1
vote
2answers
38 views

Closure of subset of affine plane

Inspired by this question, I wonder if one can prove the following Let $ k $ be an algebraically closed field. Is the closure of $ \{(x,y):x^{2}+y^{2}=1,x\ne 0\} $ in the affine plane over $ k $ ...
3
votes
1answer
49 views

Are real algebraic points dense in a real affine variety?

Let $V\subset \mathbb R^n $ be the zero-locus of finitely many polynomials with rational coefficients. Is it true that the set of points in $V$ whose coordinates are algebraic numbers is dense in the ...
2
votes
2answers
28 views

Extension of rings decreasing Krull dimension

Let $A \subset B$ a ring extension. It is well known that if the extension is integral, then $\dim B=\dim A$. I can think of some examples where the Krull dimension increases (and by that I mean $\dim ...
9
votes
0answers
93 views

Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
1
vote
1answer
45 views

Tensor power modulo cyclic group action

Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes ...
0
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0answers
28 views

which powers of maximal ideal contain/are included. the notation

Let $R$ be a (associative, commutative) local ring, denote by $\mathfrak{m}$ its maximal ideal. For any other ideal $J\subset R$ one can speak about: the biggest power $k\le\infty$ such that ...
0
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0answers
40 views

How do I prove that primary ideals satisfy this property?

Let $R$ be a commutative ring. Let $Q$ be a primary ideal of $R$. Let $I,J$ be ideals of $R$ such that $IJ\subset Q$. How do I prove that $I\subset Q$ or $J^n\subset Q$ for some positive integer ...
2
votes
1answer
39 views

Reference request for a theorem on maps to normal varieties with equidimensional fibers being open

I am requesting a reference for a proof.. I believe that it is due to Chevalley. A theorem by Chevalley says that if $f: X \rightarrow Y$ is a dominant morphism of irreducible varieties, then there is ...
0
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0answers
34 views

Inverse limits of quotient rings

Let $A\subset B$ be an extension of discrete valuation rings and let $p$ and $P$ be the non-zero prime ideals of $A$ and $B$ respectively. So I can write $pB=P^m$ for some $m>0$. I form the ...
1
vote
2answers
28 views

Radical of a ring [duplicate]

Let $A$ be a commutative ring with unity. Let the radical $\operatorname{Rad}(A)$ of $A$ be the ideal consisting of all nilpotent elements of $A$. Is $\operatorname{Rad}(A)$ of $A$ the same as ...
0
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0answers
31 views

Some confused terminology in Matsumura's textbook about completion

I am reading Cohen structure Theorem in textbook "commutative algebra" by matsumura. Here, the author repeatly mention "Assume that $A$ is a complete and separated local ring". One can easily know ...
3
votes
3answers
66 views

Flatness of $R/(x)$ with $R$ being local

Let $R$ be a commutative local ring and let $x \in R$ be a non-unit. Suppose that for all exact sequences $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ the following sequence is also ...
4
votes
0answers
113 views

When do the zero divisors of a commutative ring form an ideal?

Let $J$ denote the set of zero-divisors of a commutative ring $R$. Since we automatically have $RJ \subseteq J$, hence $J$ is automatically halfway to being an ideal. Furthermore, its already ...
0
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0answers
42 views

Grobner bases of a determinantal ideal

I've been studying algebraic geometry recently and there is a problem I'm struggling with: Suppose $A$ is a $m\times n$ complex matrix of rank $\leq r$, this is equivalent to all its $(r+1)\times ...
2
votes
2answers
54 views

Polynomial algebra and polynomial ring

What is the difference between polynomial algebra and polynomial ring? because sometimes I read polynomial algebra and it looks like a polynomial ring $K[x,y,..]$ in many variables. Thanks
0
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0answers
30 views

Having only the zero as a nilpotent element is a local property

I want to show that having only the zero as a nilpotent element is a local property for a Ring $R$. Assume $R$ only has the zero element as a nilpotent element and there exists a prime ideal $p$ ...
0
votes
1answer
32 views

An example of a c.i./Gorenstein/C.M. integral domain which is not integrally closed

If I am not wrong, it is known that: {Regular rings} $\subsetneq$ {Complete intersection rings} $\subsetneq$ {Gorenstein rings} $\subsetneq$ {Cohen-Macaulay rings}. It is known that a regular ring ...
6
votes
2answers
148 views

Does there exist such an invertible matrix?

Let $n \geq 1$ and $A = \mathbb{k}[x]$, where $\mathbb{k}$ is a field. Let $a_1, \dots, a_n \in A$ be such that $$Aa_1 + \dots + Aa_n = A.$$ Does there exist an invertible matrix $\|r_{ij}\| \in ...
0
votes
2answers
115 views

Artinian rings have finite length

In a recent question of mine here I asked whether it is true or not that Artinian (commutative) rings have finite length. I came up with a proof, and I want to know if it is valid. So, I want to ...
0
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0answers
43 views

How to understand this sentence within Atiyah-Macdonald's textbook about commutative algebra

In page 102 of this textbook, authors mentioned that: Assume topological group $G$ has a fundamental system of neighborhoods consisting of subgroups as: $G= G_0 \supseteq G_1 \supseteq\cdots\supseteq ...
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2answers
55 views

Why the dimension of $R/(a)$ is $0$?

How do I see the following fact? If $R$ has dimension $1$, and $a$ is a non-zerodivisor and non-unit, then $R/(a)$ has dimension $0$. That is saying if $P_1\supset P_2\supset (a)$ are two prime ...
0
votes
1answer
46 views

What's wrong with the following argument that every module is flat?

Okay, I know I'm doing something incredibly stupid here, but for whatever reason I can't figure out what. As I understand it, an R-module M is flat iff $f : I \otimes_R M \to I M$ is an isomorphism ...
2
votes
1answer
53 views

If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
1
vote
0answers
49 views

dimension formula for fiber product of affine varieties

Let $X \subset \mathbb{A}^n, \, Y \subset \mathbb{A}^m, \, Z \subset \mathbb{A}^{\ell}$ be irreducible affine varieties and let $f: X \rightarrow Z, \, g: Y \rightarrow Z$ be surjective morphisms. ...
0
votes
1answer
28 views

Given $A$-modules $N \subset M$ such that $N_m=M_m$ for all maximal ideals $m$, show that $M=N$

I am working on this exam question 6 $A$ is commutative ring with $1$ a) If $N \subset M$ are $A$-modules and $N_m=M_m$ for all maximal ideals $m$, show that $M=N$. We know that $N_m=M_m$ ...
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0answers
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What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
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0answers
28 views

Projective ideal of a Non-noetherian domain

If $R$ is an integral domain which is not Noetherian and let $I$ be an ideal which is not finitely generated. We have always, if I is invertible, then I is always projective. Is the converse true when ...
4
votes
2answers
61 views

Flat Non Projective $A$-Module [duplicate]

A standard fact in Commutative Algebra is that a Projective $A$-module is flat. The converse is false. Can someone show me an example of a Flat Non Projective $A$-Module? Thank you!
3
votes
1answer
38 views

Redundancy in the definition of Dedekind domain?

Is there a domain which is noetherian and whose nonzero prime ideals are maximal, but which is not integrally closed? This may be a silly question to experts. I ask because I think I have found ...
0
votes
1answer
38 views

Isomorphic Affine Schemes

If $f:A\to B$ is a homomorphism of rings such that $f':\text{spec} B \to \text{spec} A$ is a homeomorphism does it follows that the spectra are isomorphic as schemes? I was able to reduce this ...
3
votes
1answer
45 views

Local ring and isomorphism problem

I have a local ring $R$ with maximal ideal $\mathfrak{m}$. Fixing some $x\in\mathfrak{m}$, I want to show that $\mathfrak{m}^{k-1} \subset (\mathfrak{m}^k : x)$ and conclude that $R/(\mathfrak{m}^k : ...