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

learn more… | top users | synonyms (1)

0
votes
0answers
9 views

Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$

What is the Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ where $A$ is a Noetherian, commutative ring and $\mathfrak{a} = \langle f_1, \ldots, f_s \rangle$ and each $f_i$s is monic in $A$? Is ...
1
vote
0answers
29 views

Minimal Free Resolutions

Matsumura, Commutative Ring Theory, Chapter 7 p. 153-4: Let $(A, \mathfrak{m}, k)$ be a local ring. An exact sequence $$(*) \cdots \rightarrow L_i \xrightarrow{d_i} L_{i-1} ...
0
votes
2answers
53 views

What is the Krull dimension of $A[x,y,z]/\langle xy,xz \rangle$

What is the Krull dimension of $B=A[x,y,z]/\langle xy,xz \rangle$, given $A$ is a Noetherian, commutative ring? I am thinking can it be shown to be an integral extension of another ring?
0
votes
0answers
38 views

What is $\overline{D(f)}$?

Let $A$ be a ring, $f\in A$. If $A$ is Noetherian, $\text{Spec}(A)$ has finitely many irreducible components, let us call them $\{Z_i\}_{i=1}^n$. So we write $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
3
votes
0answers
45 views

What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
1
vote
1answer
17 views

$R\subset S$ rings conditions implying that there is at most $t$ maximal ideals in $S$ lying over any maximal ideal in $R$

The problem is the following. Let $R,S$ be rings such that $R\neq 0 $, $R\subset S$ and $S$ is finitely generated as a $R$-module, with $t$ generators. Let $\mathfrak{m}$ be an maximal ideal in $R$. ...
2
votes
0answers
31 views

If $G$ is shellable, then $G \backslash \{x_i\}$ is shellable?

A simplicial complex $\triangle$ is shellable on the vertex set $\{x_1,\ldots,x_n\}$, if the facets of $\triangle$ can be ordered, say $F_ 1 , . . . , F _s$ , such that for all $1 ...
3
votes
1answer
32 views

Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
2
votes
1answer
27 views

Computation of permanents of general matrices

In the following paper http://www.stat.uchicago.edu/~pmcc/reports/permanent.pdf it is stated that: "Exact computation of permanents of general matrices is a #P (sharp P) complete problem, so no ...
0
votes
0answers
7 views

How to simulate Permanental Point Process

I have simulated a determinantal point process in a square grid using Gaussian Kernel. The Gaussain matrix is decomposed into its eigenvectors and eigenvalues. In core implementation, the elementary ...
0
votes
3answers
54 views

Domain strictly contained in the intersection of localizations at the 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 ...
0
votes
1answer
23 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 ...
5
votes
2answers
74 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
64 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
35 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
votes
1answer
40 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
votes
2answers
55 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
43 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
58 views

when is the cokernel 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? ...
4
votes
1answer
70 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
63 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.
0
votes
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
68 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
73 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
53 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
39 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
votes
0answers
65 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?
4
votes
0answers
69 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
votes
1answer
57 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
49 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
votes
0answers
90 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
votes
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
votes
0answers
37 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 ...