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

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6
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2answers
222 views

Question on Noetherian/Artinian properties of a graded ring

Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the ...
6
votes
1answer
252 views

Equivalent condition for flatness of an A-module (Atiyah-MacDonald ex. 2.26)

I would like to solve the following exercise (2.26) from Atiyah & MacDonald's "Introduction to Commutative Algebra": If $M$ is an $A$-module (where $A$ is a commutative ring), then: $$M \text{ is ...
6
votes
1answer
236 views

Can this be salvaged to give a proof that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}(\sqrt[3]{2})$?

Recently, I was intrigued by the question asking for an easy way to show $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}(\sqrt[3]{2})$. I was playing with the approach, trying to ...
6
votes
1answer
224 views

Modules $M$ such that the automorphism of $M \otimes M \otimes M$ induced by the permutation $(123)$ is the identity

I've been struggling with the following problem for a couple of days and I don't seem to get any further: Let $R$ be a commutative ring. I would like to get (something like) a classification of all ...
6
votes
1answer
223 views

inverse limit of isomorphic vector spaces

Let $$\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $$ be an inverse system of finite dimensional vector spaces with the property that the $A_i$ are ...
5
votes
1answer
37 views

Example (for some $X$) of a nonclosed ideal in $\mathbb{C}(X)$?

Let $X$ be a compact metric space and $\mathbb{C}(X)$ the algebra of continuous functions $f: X \to \mathbb{C}$, with pointwise operations. We equip $\mathbb{C}(X)$ with the maximum norm $N(f) := ...
5
votes
1answer
129 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
5
votes
1answer
78 views

Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My ...
5
votes
2answers
140 views

Ring with maximal ideal not containing a specific expression

Main question : May there exist an integral domain $R$, with fraction field $K$, that fulfills the following condition: there exists $x\in K$, $x\not \in R$ and a maximal ideal $\frak m$ of $R[x]$, ...
5
votes
1answer
159 views

Can we have a Primary Avoidance Theorem ?

Prime Avoidance Theorem says: Let $ P_1, P_2,\dots, P_n $ be prime ideals in a commutative ring $R$ and let $I$ be an ideal of $R$ such that $ I \subseteq P_1 \cup P_2 \cup \cdots \cup P_n$. ...
5
votes
2answers
364 views

Is every field the field of fractions for some integral domain?

Given an integral domain $R$, one can construct its field of fractions (or quotients) $\operatorname{Quot}(R)$ which is of course a field. Does every field arise in this way? That is: Given a ...
5
votes
2answers
215 views

a flatness criterion

I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
5
votes
1answer
273 views

Generalizing Artin's theorem on independence of characters

Artin's theorem says that for any field $K$ and any (semi) group $G$, the set of homomorphisms from $G$ into the multiplicative group $K^*$ is linearly independent over $K$. Can this theorem be ...
5
votes
2answers
266 views

Localization Notation in Hartshorne

This is a question about notation in Hartshorne's Algebraic Geometry. According to my understanding $k[x_0,\cdots,x_n]_{(x_i)}$, (see e.g. page 18), consists of the elements of degree zero in the ...
5
votes
2answers
206 views

When does $\mathfrak{a}B\cap A = \mathfrak{a}$?

Let $A\subset B$ be rings, and let $\mathfrak{a}$ be an ideal of $A$. Under what circumstances does $\mathfrak{a}B\cap A = \mathfrak{a}$? More precisely, are there conditions on $A,B$ that guarantee ...
5
votes
1answer
360 views

Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?

This question is related to the one here: A question related to krull akizuki In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily ...
5
votes
1answer
2k views

A good commutative algebra book [duplicate]

Possible Duplicate: Reference request: introduction to commutative algebra I'm looking for a good book on commutative algebra covering most of (but not limited to) : Basic Galois theory ...
5
votes
1answer
354 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
5
votes
1answer
435 views

The ring of germs of functions $C^\infty (M)$

Define $C^\infty (M)_x := \{ (U,f) | x \in U $ open $ , f \in C^\infty (U) \} / \sim $ where $M$ is a manifold and $(U,f) \sim (V,g)$ if $\exists W$ open, $x \in W$ such that $W \subset V \cap U$ ...
4
votes
2answers
92 views

Power set representation of a boolean ring/algebra

Let $R$ be a finite boolean ring. It's known that there's a boolean algebra/ring isomorphism $R\cong \mathcal P(\mathsf{Bool}(R,\mathbb Z_2))$. I'm trying to get a feel for this. The subsets of ...
4
votes
0answers
110 views

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $M ...
4
votes
1answer
59 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 ...
4
votes
2answers
196 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are ...
4
votes
0answers
157 views

Why is the completion of the ring of germs of smooth functions $\cong \mathbb{R}[|T|]$?

Let $C^{\infty}$ be the canonical commutative ring on the set $\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ smooth}\}$. Let $\mathfrak{m}= \{ f \mid f(0)=0 \}$ a maximal ideal. Consider the ...
4
votes
2answers
132 views

$Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$

Let $(I:J)$ denote the colon ideal (or ideal quotient). It is pretty clear that the Zariski closure of $Z(I)-Z(J)$ is contained in $Z(I:J)$. How can we prove that the the Zariski closure of ...
4
votes
2answers
162 views

Surjection from a Noetherian ring induces open map on spectra?

Let $A$ be a Noetherian ring, $f: A\rightarrow B$ a surjective ring map, then should the induced map on spectra $f^*: Spec(B)\rightarrow Spec(A)$ be an open map? In Atiyah and Macdonald, Chapter 1, ...
4
votes
2answers
227 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
4
votes
1answer
456 views

Finitely generated modules over noetherian rings isomorphic to their double duals

Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module. Suppose that $M$ is isomorphic to the double dual, how can I prove that $M$ is reflexive? (i.e. it is isomorphic to the double ...
4
votes
1answer
305 views

The completion of a noetherian local ring is a complete local ring

We have defined the completion of a noetherian local ring $A$ to be $$\hat{A}=\left\{(a_1,a_2,\ldots)\in\prod_{i=1}^\infty A/\mathfrak{m}^i:a_j\equiv a_i\bmod{\mathfrak{m}^i} \,\,\forall ...
4
votes
1answer
771 views

Understanding the conductor ideal of a ring.

Consider the inclusion of a ring $A$ into its integral closure $B$. The conductor ideal $I$ is defined as $I:=\{a\in A~|~aB\subseteq A\}$. This is supposed to describe the locus where the ...
4
votes
1answer
222 views

Ascending chain conditions on homogeneous ideals

Here is one exercise from some notes on graded rings. I tried but I got no idea to solve it. Please help me. Thanks. Let $R$ be a graded ring. Prove that $R$ is Noetherian (Artinian) if and only ...
4
votes
2answers
845 views

Example of a non-Noetherian complete local ring

I was looking for an example of a non-Noetherian complete local commutative ring with $1$. I would appreciate if anyone can point to a reference.
4
votes
3answers
2k views

irreducibility of a polynomial in several variables over ANY field

The irreeducibility of a polynomial $f\!\in\!K[x_1,\ldots,x_n]$ in general depends on what the field $K$ is (for example, if $K=\mathbb{R}$, then $f=x_1^2+1$ is irreducible, but if $K=\mathbb{C}$, it ...
4
votes
2answers
426 views

Every ideal of $K[x_1,\ldots,x_n]$ has $\leq n$ generators?

Is this true: Every ideal of $K[x_1,\ldots,x_n]$ is generated by some subset with $\leq n$ elements? It is true when $n=1$, since $K[x]$ is a PID. I'm trying to prove it is not true for $n\geq2$, ...
4
votes
1answer
600 views

generators of a prime ideal in a noetherian ring

Suppose $R$ is a Noetherian ring and $P$ is a prime ideal. If the number of generators of $PR_P$ as an ideal in $R_P$ is $n$, can we say anything about the number of generators of $P$ as an ideal of ...
3
votes
2answers
90 views

Radical ideal of $\langle x^2+y^2+z^2, xy+yz+xz\rangle$

The following is exercise 3.7 from Undergraduate Algebraic Geometry by Reid. Let $J=\langle x^2+y^2+z^2, xy+yz+xz\rangle$; identify $V(J)$ and $I(V(J))$. The question did not specify the field. ...
3
votes
2answers
107 views

Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
3
votes
3answers
278 views

Regular Ring is Integrally Closed?

Studying some topics in Algebraic Geometry I've bumped into the following question: Let $A$ be a regular ring. Is $A$ integrally closed? Someone said me that with the hypothesis $A$ local ...
3
votes
1answer
116 views

on the proof of Theorem 4.3.2 in Bruns & Herzog ``Cohen-Macaulay Rings" (Gotzmann's regularity theorem)

The theorem and the first part of its proof is shown below: In particular, the authors conclude (2 lines below equation (2)) that $(i): P_R(n) = {n + a_1 \choose a_1}+\cdots+ {n+a_r -(r-1) \choose ...
3
votes
1answer
123 views

When a holomorphy ring is a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
3
votes
1answer
264 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
3
votes
1answer
103 views

Is Serre's $S_1$ condition equivalent to having no embedded primes?

Today I tried to prove that if a Noetherian ring $A$ satisfies Serre's $R_0$ and $S_1$ conditions, then $A$ is reduced. Now we recall that $R_0$ means the localization at any minimal prime is a field ...
3
votes
1answer
94 views

Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated? I feel like the example (or proof of impossibility) ought to be obvious, but I'm not seeing it.
3
votes
3answers
142 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
3
votes
2answers
222 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
3
votes
0answers
107 views

When does the inverse limit preserve the localisation?

Question When is the following true? $$\varprojlim(S_\alpha^{-1}A_\alpha)\cong(\varprojlim S_\alpha)^{-1}(\varprojlim A_\alpha)$$ (For details, consult the next part.) Notations One can ...
3
votes
2answers
184 views

Associated primes of a quotient module.

Let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module and $p\in \operatorname{Ass}(M)$. Suppose $x$ is an $M$-regular element and $q$ is a minimal prime over $I=(p,x)$. How can we show ...
3
votes
3answers
401 views

Commutative integral domain does not finitely generate its field of fractions

I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that ...
3
votes
0answers
437 views

Quick question on localization of tensor products

All rings are commutative with unit. Let $\rho:A\rightarrow B$ be a ring homomorphism. Suppose $\mathfrak q$ is a prime ideal of $B$, and let $\mathfrak p=\rho^{-1}(\mathfrak q)$. My question: Is ...
3
votes
2answers
189 views

Ideal of polynomials in $k[X_1,…X_n]$ vanishing at a point $p$ is $(X_1 - p_1, …,X_n - p_n)$ [duplicate]

I'm having a little trouble following Eisenbud here: My problem is that I don't see how the isomorphism $${k[x_1,...,x_n] \over \mathfrak{m}_p} \cong k$$ is constructed. This seems a bit ...