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

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2
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3answers
131 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
30 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
52 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
45 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
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0answers
65 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
39 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 ...
2
votes
0answers
166 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
75 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 ...
0
votes
0answers
131 views

Integral closures of algebras of finite type over the ring of algebraic integers of an algebraic number field

Let $K$ an algebraic number field, i.e. a finite extension of $\mathbb{Q}$. Let $A$ be the ring of algebraic integers of $K$, $B$ an algebra of finite type over $A$ without zero-divisors, $B'$ ...
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
102 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
30 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
105 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
41 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
49 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
39 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?
1
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0answers
39 views

Comultiplication in graded Hopf Algebras

Let $H$ be a graded Hopf algebra over some commutative ring $k$. I'm looking for a proof of the following result, which seems to be stated in various locations. For $x$ in $H$ of degree $n$ ...
1
vote
1answer
32 views

Finding a specific module.

I want to think of a module $M$ over a commutative ring with identity $R$ such that $M \oplus N = R^3 $ while $N$ is isomorphic to $M$. Are there some interesting examples which satisfies this ...
1
vote
2answers
49 views

A question on a problem on localization from Atiyah (3.8)

I was having trouble with the following problem from Chapter $3$ of Atiyah-MacDonald Let $S, T$ be multiplicatively closed sets in the ring $A$, such that $S\subseteq T$. Let $\varphi : S^{−1}A \to ...
1
vote
2answers
51 views

Why rings of fraction is defined as that?

Let $R$ be the commutative ring, S be the multiplicative set. As we know in $S^{-1}R$, $a/b\sim c/d$ iff there is a $t \in S$ such that $adt=bct$. The question is that why we need $t$? Why not just ...
1
vote
1answer
46 views

Basic open sets in the Zariski topology are also compact.

Let $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complements ...
1
vote
1answer
23 views

Extending and contracting an ideal by a faithfully flat homomorphism

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $. This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...
0
votes
1answer
50 views

Suppose that $R$ is a one-dimensional normal Noetherian local ring. Then the maximal ideal $m_R$ is principal

Theorem 11.2 (Matsumura's Commutative Ring theory) gives us equivalent conditions for a ring $R$ to be considered a DVR. I was stuck while reading the proof of $(4) \implies (3)$, (3) $R$ is a ...
0
votes
0answers
37 views

Intersection of modules is equal to product.

If $B$ is a commutative ring and let $\mathcal{Q}_1,\ldots,\mathcal{Q}_n$ ideals relative primes. Let $M$ be a $R$-module. I don't sure if this is true. Then $$(\mathcal{Q}_1\cap\cdots ...
0
votes
0answers
26 views

Describe the differential of $d\phi : \mathbb{T}_{t, \mathbb{A^1}} \rightarrow \mathbb{T}_{t^3, t^4, t^5, W}$

Let $V = \mathbb{A^1}$ and $W = Z(xz - y^2, yz, x^3, z^2 - x^2y) \subset \mathbb{A^3}$ and let $\phi: V \rightarrow W$ be a surjective morphism, describe the differential. Currently in a course in ...
2
votes
1answer
124 views

Finding a finite generating set of an ideal of monomials

My problem involves considering the ideal $I = \{ X^mY^n \mid m,n\in \mathbb{N}, m^2n>5 \}$ of $\mathbb{Q}[X, Y]$. I am asked to write down a finite generating set of $I$ and explain how I ...
0
votes
1answer
35 views

Want to show that $g\in I$ where $I$ is an ideal, given the following conditions

Let $R=K[x_1,...,x_n]$ and $I$ be an ideal of $R$, $K$ being a field Given $h\in I$, $g\in \sqrt{I}$ and $f\in\sqrt{I}$ Where $in_<(f)=in_<(h)$ and $g=f-h$. So $in_<(g) < ...
0
votes
2answers
31 views

Relation between Variety of $(I\cap J)$ and Variety of $(I)$ $\cap$ Variety of $(J)$

I was wondering whether a relationship exists between $V(I\cap J)$ and $V(I)\cap V(J)$. Where $I$ and $J$ are ideals of the ring $R=K[x_1,...,x_n]$.
2
votes
1answer
81 views

Is the intersection of two Noetherian rings Noetherian?

Is the intersection of two Noetherian rings also Noetherian? If yes, could you please give me the idea of proof. If not, give me an counterexample.
1
vote
1answer
27 views

$\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$

I'm having trouble trying to show that $\mathbb{Q}[t]$ is integrally closed in $Quot(\mathbb{Q}[t])$. Where $Quot(\mathbb{Q}[t])$ is the field of fractions of $\mathbb{Q}[t]$. So I'm trying to show ...
1
vote
1answer
32 views

Line Bundles on Local Rings

Let $A$ be a local ring and $L$ a module over $A$ which is projective and of rank one. Does it follow that $L$ is isomorphic to $A$?
0
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1answer
38 views

Finding $\mathbb{Z}[\sqrt{-3}]/(p)$ for some prime $p$.

I have to prove that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_{p^{2}}$ if $p\equiv 5\ \text{mod}\ 6$ and $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_{p}\oplus\mathbb{F}_{p}$ if $p\equiv 1\ ...
0
votes
1answer
25 views

Showing that $in_<(f^m) = in_<(f)^m$

I am currently in the following scenario: Let $I$ be an ideal of $K[x_1, ..., x_n]$, $<$ be a fixed term order and $in_<(I)$ be radical. I want to show that: $in_<(f^m) = in_<(f)^m$ ...
2
votes
1answer
36 views

Proof of Unique factorization in Dedekind Rings .

Proof de unique factorizaation in Dedekind Rings. Algebraic Number fields, Janusz, Second edition. In the above proof, Theorem 3.13. Why of the corolary 3.7, ...
0
votes
1answer
72 views

“Going between” property

Let $A \subset B$ be an integral ring extension and assume that $A$ is a finitely generated $K$-algebra over some field $K$. Let $P_1\subsetneq P_3$ be prime ideals of $A$ and let $Q_1\subsetneq ...
-1
votes
1answer
60 views

Counterexample to the finiteness of integral closure of a Dedekind domain.

Let A be a Dedekind domain, K its field of fractions, L/K a finite extension, B the integral closure of A in L. By the Krull-Akizuki theorem, B is noetherian, hence B is a Dedekind domain. In the ...
2
votes
2answers
115 views

$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$

I know this question has been asked before, but I think I'm very close to a new solution and wanted to know if it is a viable approach. Let $C(X)$ be the ring of continuous functions $X \rightarrow ...
4
votes
3answers
177 views

``Minimal generating ring" for a field of fractions

In this answer and the linked MathOverflow post, it's shown that any field $F$ of characteristic zero contains a proper subring $A$ such that $F$ is the field of fractions of $A$. However, there is ...
2
votes
1answer
38 views

Viewing the universal property of rings of fractions as a universal arrow

For a multiplicatively closed subset $S$ of $A$, we have a functor $S^{-1}: A-Mod \rightarrow S^{-1}A-Mod$. I am trying to understand this functor a little bit better and I was thinking about the ...
0
votes
1answer
45 views

How to see $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$ as a ring?

Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the ...
0
votes
1answer
31 views

$A\subset B $ with $B$ integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$?

Let $A\subset B$ with $B$ an integral domain. If $B$ is integral over $A$ can we say that $Q(B)$ is algebraic over $Q(A)$ ? (Here $Q(\dots)$ denotes the quotient field of $(\dots))$.)