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

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0
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1answer
28 views

Domain strictly contained in localization of primes of height one

If $R$ is a normal domain, then it is equal to the intersections of localizations at height one primes. Maybe Noetherian is required here, I'm more than willing to assume it. What is an example of a ...
0
votes
1answer
21 views

Non-finitely generated, non-divisible, non-projective, flat module, over a polynomial ring

(1) Let $R=k[x_1,\ldots,x_n]$. I wish to find an example of a non-finitely generated, non-divisible, non-projective, flat $R$-module. Notice that $k(x_1,\ldots,x_n)$ is NOT an example of what I am ...
4
votes
2answers
58 views

Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
2
votes
0answers
61 views

Fields of Research in Algebra [on hold]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
3
votes
1answer
33 views

Global Dimension of a Ring and its Localizations

Why is the following true? The global dimension of a noetherian ring $A$ is the supremum of the global dimension at its localizations at its maximal ideals: ...
3
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1answer
38 views

Is it true that every prime ideal of height one is principal? [on hold]

Is it true that every prime ideal of height one is principal ? Please help
3
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2answers
47 views

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
3
votes
1answer
39 views

Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat ...
4
votes
2answers
53 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
3
votes
1answer
49 views

when is the quotient of a map of free modules free?

Let $R$ be a commutative ring (noetherian if needed) and $n,m$ be two nonnegative integers. Consider a map $\varphi: R^n\rightarrow R^m$ Is there a characterisation, e.g. in terms of the matrix ...
3
votes
1answer
32 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? ...
3
votes
1answer
64 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
62 views

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

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.
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2answers
65 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
65 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
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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
72 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
50 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 ...
0
votes
0answers
38 views

When is the Zariski closure of subset connected [closed]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is an indecomposable ring, how can we show that the ...
2
votes
1answer
37 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
87 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 ...
0
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0answers
62 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
43 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, ...
-1
votes
0answers
39 views

Question about Poincare series

Let $R=\mathbb Q[x,y]_{(x,y)}$ and $I=(x^{10},x^8y,xy^4,y^5)$. Then how can we calculate the Poincare series of $I$ by Macaulay 2?
0
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0answers
49 views

When is a homomorphic image of a polynomial ring self injective [closed]

Is the ring $\dfrac{\mathbb{Z}_3[x, y]}{(x^2y)}$ self injective?
4
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0answers
68 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
45 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
49 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
76 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
16 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 ...
0
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1answer
56 views

Module of constant rank over noetherian reduced ring

Let $A$ be a reduced noetherian commutative ring and $M$ be a finitely-generated $A$-module such that for all prime ideals $\mathfrak p$, $M_{\mathfrak p}/\mathfrak pM_{\mathfrak p}$ is an ...
2
votes
3answers
75 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
48 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 ...
2
votes
0answers
50 views

On why $k(X)^{G}$ is a finitely generated field extension

In a book I was reading, from the assumptions that we have a linear algebraic group $G$ acting on an irreducible (affine) variety $X$, the author writes that $k(X)^{G}$ is a finitely generated field ...
1
vote
2answers
36 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
26 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 ...
8
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0answers
89 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 ...
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0answers
26 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
36 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
38 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
33 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
26 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
65 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
112 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
votes
0answers
39 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
50 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
votes
0answers
29 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 $p$ ...