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

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12
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1answer
107 views

Is there an algebraic non-rational extension of the integers, whose set of prime elements contains the prime integers?

Let the ring $\mathbb{Z}[\alpha]$ with $\alpha$ an algebraic number. Let $P(\mathbb{Z}[\alpha])$ be the set of all the prime elements of $\mathbb{Z}[\alpha]$. Question: Is there $\alpha$ algebraic ...
1
vote
1answer
42 views

Large product of matrices equal zero but not small ones

Let $\cal F$ be a finite set of matrices in the same ${\cal M}_n({\mathbb R})$ for some fixed $n$. Following the ideas in that recent MSE question, let us say that $\cal F$ is jointly nilpotent if ...
0
votes
1answer
48 views

$(1-\zeta_m)$ is a unit in $\mathbb{Z}[\zeta_m]$ if m contains at least two prime factors

We know that for $m=p^r, 1-\zeta_m$ is a prime.Now suppose that m has at least 2 distinct primes appearing in its prime factorization,we need to show that $1-\zeta_m$ is a unit in its ring of integers ...
5
votes
1answer
71 views

Is a square of a prime ideal in a UFD always primary?

More concretely, Let $R$ be a UFD and $\mathfrak{p}$ a prime ideal ideal of $R$. Does it always hold that $\mathfrak{p}^2$ is a primary ideal? I know that it always holds if $\mathfrak{p}$ is a ...
0
votes
1answer
55 views

Ideals lying over a given prime ideal

For the following two ring extensions $R\subset S$: 1) $\mathbb{R}[y]\subset \mathbb{R}[x,y]/(x^2-y);$ 2) $\mathbb{Z}\subset\mathbb{Z}[x]/(x^2-3)$ find in each case, i) a nonzero prime ideal $P$ ...
0
votes
1answer
74 views

Inseparably ramified primes over $\mathbb{Z}[x]$

Let $A =\mathbb{Z[x]}$ be the ring of polynomials of one variable over $\mathbb{Z}$. Let $K$ be its field of fractions, $L/K$ be a finite extension, $B$ the integral closure of $A$. Since $A$ is a ...
4
votes
0answers
49 views

Question about Proj of graded ring

Let $A=k[x_0, x_1,x_2]$ be a polynomial ring such that $\text{deg}(x_0)=2$, $\text{deg}(x_1)=\text{deg}(x_2)=1$. How can I understand what is $X=\text{Proj}\,A$?
0
votes
1answer
67 views

Finding maximal ideals and Krull dimension

I have difficulties in finding the maximal ideals and compute the dimension of a quotient ring $\mathbb{C}[x,y,z]/(x^2-y^2,z^2x-z^2y)$. Here $(x^2-y^2,z^2x-z^2y)$ is a product of $(x+y,z^2)$ and ...
2
votes
0answers
50 views

Ramification theory on Noetherian integrally closed domains

Let $A$ be a Noetherian integrally closed domain, $K$ its field of fractions. For a non-zero ideal $I$ of $A$, we write $I^{-1} = \{x \in K | xI \subset A \}$. If $I^{-1} = J^{-1}$, we wrire $I \cong ...
2
votes
1answer
55 views

Stable filtrations

An $\mathfrak{a}$-stable filtration is (as many know) an $\mathfrak{a}$-filtration $\{M_n\}$ such that for large $n$; $\mathfrak{a}M_n=M_{n+1}$. This is saying in some sort of way (I think) that the ...
2
votes
2answers
47 views

Structure theorems for modules over 'good' rings

Structure theorem for finitely-generated modules over PID is well-known fact. But is there similar theorems for modules(maybe finitely-generated) over noetherian or artin or some other 'good' rings? I ...
1
vote
2answers
95 views

Dimension of local rings on scheme of finite type over a field.

In chapter III Hartshorne seems to be using without proof or mention a theorem on the dimensions of local rings of schemes of finite type over a field. I know that for an integral scheme of finite ...
5
votes
3answers
198 views

Example of a finitely generated faithful torsion module over a commutative ring

Can a finitely generated module $M$ over a commutative ring have $\operatorname{Ann}(x) \neq 0$ for all $x \in M$ while $\operatorname{Ann}(M) = 0$? It's not difficult to show that there is no such ...
1
vote
0answers
38 views

Morphism between projective schemes induced by injection of graded rings

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings ...
0
votes
2answers
33 views

Example to show that multiplication by ideals and intersection of submodules do not commute

The key point of question about typical proof of Krull Intersection Theorem is that multiplication by ideals and intersection of submodules do not commute. Can anyone give me an example of this? ...
0
votes
1answer
43 views

$((I \cap J)^{-1})^{-1} = (I^{-1})^{-1} \cap (J^{-1})^{-1}$ for ideals of an integral domain?

Let $A$ be an integral domain and $I$, $J$ be non-zero ideals. Is $((I \cap J)^{-1})^{-1} = (I^{-1})^{-1} \cap (J^{-1})^{-1}$? For an ideal $I$, we define $I^{-1} = \{x \in K | xI \subset A\}$, ...
1
vote
1answer
66 views

Problem 2.26 in Fulton's Algebraic Curves: redundant hypothesis?

The problem reads: "Let $R$ and $S$ be DVRs with maximal ideals $M = (q)$ and $N = (p)$ respectively, $K$ the quotient field of $R$. Suppose $R \subset S \subset K$, and suppose that $M \subset N$. ...
1
vote
1answer
46 views

A question about the proof of the map from Spec B to Spec A is continuous

Here, $f:A \to B$ is a ring homomorphism, thus to any $Q\in\textrm{Spec}( B)$, $f^{-1}(Q)$ is in $\textrm{Spec}( A)$, which induces the map $g: \textrm{Spec}( B) \to \textrm{Spec}( A)$ by Sending $Q$ ...
1
vote
1answer
67 views

Intersection of submodules

I have question regarding intersection of submodules. Could anyone give example of a commutative ring $R$ with identity and an $R$-module $M$ such that $$IM\cap JM\nsubseteq (I\cap J)M$$ for some ...
1
vote
0answers
26 views

Hilbert series of exterior algebra?

Let the exterior polynomial algebra $A=\Lambda_K[x_1,\ldots,x_n]$ have grading $\deg x_i=d_i$. Is there some nice formula for the Hilbert-Poincare series $HP_A=\sum_k\dim_K ...
0
votes
0answers
48 views

Van der Waerden's ideal theory on Noetherian integrally closed domains

In his book Algebra, he developed an ideal theory of Noetherian integrally closed domains(the section 105). He generalized the ideal theory of Dedekind domains on these rings. Near the end of the ...
3
votes
1answer
65 views

Show an ideal is a finitely generated projective module via a split exact sequence

Let $I$ be an ideal of $R$ such that the mapping $f:I\otimes_R\operatorname{Hom}_R (I,R)→R$ defined (on the generators) by $f(i\otimes α)=α(i)$ for all $i∈I$ and $α∈\operatorname{Hom}_R (I,R)$ is ...
2
votes
0answers
192 views

Why do the map from elements to sum is surjective?

Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a ...
1
vote
1answer
43 views

What is the quotient field of a localization of a domain

If $R$ is an integral domain, $K=\text{Frac}(R)$ and $S$ is a multiplicative subset of $R$ then is $K$ also the quotient field of $R_S$? I've tried showing that the ring map $K \to ...
2
votes
3answers
102 views

Does a long exact sequence of flat modules remain exact after tensoring with an arbitrary module?

In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking ...
0
votes
0answers
30 views

Can we show , without considering the real numbers , that $\mathcal N$ is a maximal ideal of $\mathcal C$?

Let $\mathcal C :=\{(r_n)\subseteq \mathbb Q : \forall k \in \mathbb Q^+ , \exists N_k \in \mathbb N : |r_n-r_m| < \dfrac1{k} , \forall n,m \ge N_k \}$ and $\mathcal N:=\{(r_n)\subseteq \mathbb Q ...
2
votes
3answers
137 views

Is a height one ideal in a UFD principal?

One of the defining features of a UFD is that any height one prime ideal is principal (see Wikipedia). Is it also true that any height one (i.e. every prime minimal among those containing it has ...
1
vote
0answers
31 views

The prime meadow of a meadow

Let $(R,(-)^{-1})$ be a meadow, i.e. $R$ is a commutative ring and $(-)^{-1}$ is a unary operation on the underlying set of $R$ satisfying $(x^{-1})^{-1} = x$ and $x \cdot x^{-1} \cdot x = x$ for all ...
1
vote
1answer
53 views

Affinization of a normal variety

By affinization of $X$ I mean $\text{Aff}(X) := \text{Spec}(\Gamma(X, \mathcal{O}_X))$. First, I claim that if $X$ is reduced, then $\text{Aff}(X)$ is reduced. The argument goes: if $\Gamma(X, O_X)$ ...
1
vote
1answer
68 views

Criterion for Irreducible Monomial Ideals

I am working on the following problem: Show that (a monomial ideal $I \subset K[x_1, ... , x_n]$ cannot be written as the intersection of two strictly larger monomial ideals) if only if (I has a ...
-1
votes
1answer
44 views

Projective dimension of module over local ring

This question arose reading the well known article by Buchsbaum Lectures on regular local rings. He states without proof that, given $(R,m)$ a local ring and an $R$-module $M$ over $R$, we have the ...
1
vote
0answers
68 views

Generated by global sections vs generated in degree zero

Let $\mathcal{F}$ be a sheaf on $\mathbb{P}_k^n$ and consider the graded $k[x_0,\ldots,x_n]$-module $M=\bigoplus_{j \geq 0}\textrm{H}^0(\mathbb{P}_k^n, \mathcal{F}(j))$ ($k$ is a field). Can you give ...
1
vote
1answer
42 views

An inverse limit exact sequence for complete modules

Let $A$ be a commutative complete ring with unit for the $I$-adic topology, where $I$ is the ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such that $I^{n+1}M_n=0$ and that there exist a ...
10
votes
0answers
219 views

Two discrete valuation rings one of which is contained in another

Let $A$ and $B$ be discrete valuation rings of the same field of fractions. Suppose $A \subset B$. Then $A = B$? I came up with this problem when I was reading van der Waerden's Algebra. The ...
3
votes
1answer
76 views

A stronger definition of locally free modules

Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, Section 4.6, Exercise 4.12 (a) tells us if $M$ is a finitely presented $R$-module, then $M$ is projective if and only if $M$ is ...
1
vote
1answer
55 views

Application of Krull's principal ideal theorem

Let n be a positive integer, and let $P_0\subsetneq P_1\subsetneq ...\subsetneq P_n$ be a chain of prime ideals in a Noetherian ring R. Moreover, let $a\in P_n$. Prove: 1.There is a chain of prime ...
0
votes
1answer
44 views

Faithfully Flat Abelian Groups

I need some help to find faithfully flat abelian groups. Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is ...
0
votes
1answer
38 views

What is $I(\{(0,0)\})$?

I have two (algebraic) sets: $X_1 = Z(x) \subseteq \mathbb{A}^2$, ie, $X_1 = \{(0,y):y \in \mathbb{K}\} \subseteq \mathbb{A}^2$ $X_2 = Z(x+y^2) \subseteq \mathbb{A}^2$, ie, $X_2 = \{(-y^2,y):y \in ...
0
votes
2answers
103 views

$\mathbb{Z}[x_{1},\dots,x_{n}]/I$ is a field therefore it's finite [duplicate]

I'd spent much time for this but didn't get any results.. Could u give me only the idea but not a full proof
0
votes
1answer
27 views

Finitely generated as an Algebra

Let $R,S$ be rings. Is the following equivalent to saying $S$ is finitely generated as an $R$-algebra? "For some $n \in \mathbb{N} $ there exists a surjective ring homomorphism from ...
1
vote
1answer
31 views

Show that the ring $A(U)=A_f$ depends only on $U$ and not on $f$.

Let $A$ be a ring and let $X=Spec(A)$ and let $U$ be a basic open set in $X$. (i.e. $U=X_f$ for some $f∈A$). If $U=X_f$, show that the ring $A(U)=A_f$ depends only on $U$ and not on $f$. My Work: ...
5
votes
0answers
118 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
4
votes
1answer
38 views

Ring of continuous functions is integral over a subring

Is the ring of all continuous functions $\mathbb{R}^2 \to \mathbb{R}$ integral over the subring of functions $f$ such that $f(1,0) = f(0,1)$?
2
votes
1answer
35 views

Spectrum of Cohen-Macaulay rings and vanishing of sections

Let $R$ be a Noetherian Cohen-Macaulay ring and $X:=\mathrm{Spec}(R)$. Let $r \in R$ be an element which vanishes on an open dense set of $X$. Is it true that $r=0$?
2
votes
1answer
45 views

divisible modules over Dedekind Domains

L. Fuchs in one of his articles says that: "divisible modules over Dedekind Domains can be completely characterized by numerical invariants". Please introduce me to a source in this respect. I so ...
0
votes
0answers
27 views

Embedding a ring in a direct product

If an $R$-module $C$ is a homomorphic image of a direct sum $⊕M$, where $M$ is an $R$-module, and $R$ could be embedded in a direct product $ΠC$, could $R$ be embedded in a direct product $ΠM$?
1
vote
1answer
52 views

Proving an inclusion related to algebraic sets and interpreting it

I want to prove that $I(X_1 \cap X_2) = \sqrt{I(X_1)+I(X_2)}$ for algebraic sets $X_1=Z(G_1)$ and $X_2=Z(G_2)$, with $G_1,G_2 \subseteq \mathbb{K}[X_1,\ldots,X_n]$. Remark: Unfortunately I ...
1
vote
1answer
38 views

Characterization of prime ideals of $S^{-1}R$ when $S=1+I$, $I$ an ideal?

How can we characterize the prime ideals of $S^{-1}R$ when $S=1+I$, and $I$ is an ideal? Clearly if $p$ is a prime containing $I$ then $S^{-1}p$ is a prime of $S^{-1}R$
1
vote
3answers
50 views

Prime and Maximal Ideals

I have proved that $<x>$ is a prime but not maximal ideal in $\mathbb{Z}$[x]. I am asked to prove I is maximal in $\mathbb{Z}$[x]. $\\$ I = {$f$ $\in$ $\mathbb{Z}$[x] : the constant term of $f$ ...
0
votes
1answer
40 views

An ideal with homogeneous radical is homogeneous

Let $I$ be an ideal of a graded ring $A$. Is it possible that $rad(I)$ is an homogeneous ideal of $A$, but $I$ is not homogeneous?