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

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11
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2answers
321 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
11
votes
1answer
168 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
11
votes
1answer
419 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
11
votes
2answers
2k views

Show $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain

I'm attempting to modify the proof the $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain to prove a similar result for $\mathbb{Z}[\sqrt{6}]$. The idea is to prove that $\mathbb{Q}[\sqrt{6}]$ is Euclidean ...
11
votes
1answer
242 views

When some polynomials in $\mathbb Z[X]$ determine a regular sequence in $\mathbb Z[X_1,\dots,X_n]$?

Let $f_1,\dots,f_n\in\mathbb Z[X]$ be non-constant polynomials (not necessarily distinct). Is it true that $f_1(X_1),\dots,f_n(X_n)$ is a regular sequence in $\mathbb Z[X_1,\dots,X_n]$? The ...
11
votes
1answer
559 views

Atiyah-MacDonald help with exercise 5.10

This is an exercise from Atiyah-MacDonald, if someone can give an idea on how to prove that $a)\Rightarrow b)$: Let $f:A\rightarrow B$ a ring homomorphism. a) ...
10
votes
5answers
1k views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
10
votes
3answers
373 views

Number of prime ideals of a ring

Could anyone tell me how to find the number of distinct prime ideals of the ring $$\mathbb{Q}[x]/\langle x^m-1\rangle,$$ where $m$ is a positive integer say $4$, or $5$? What result/results I need to ...
10
votes
1answer
645 views

Prove that $\mathbb{C}[x,y] \ncong \mathbb{C}[x]\oplus\mathbb{C}[y]$

Prove that $\mathbb{C}[x,y] \ncong \mathbb{C}[x]\oplus\mathbb{C}[y]$ $\mathbb{C}[x,y]$ is the polynomial ring of two variables over $\mathbb{C}$. I guess that we can consider images of $xy$ and ...
10
votes
3answers
1k views

Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
10
votes
4answers
7k views

A ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
10
votes
2answers
1k views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
10
votes
2answers
558 views

A non-nilpotent formal power series with nilpotent coefficients

Does anyone have an example of a formal power series $$p=a_0+a_1x+ a_2x^2 + \cdots \in R[[x]]$$ ($R$ is a commutative ring) all of whose coefficients $a_i$ are nilpotent in $R$ such that $p$ is not ...
10
votes
3answers
2k views

What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
10
votes
2answers
2k views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
10
votes
2answers
234 views

Is $k[x,y,z]/(x^2+y^2-z^2)$ a UFD?

Let $k$ be an algebraically-closed field of characteristic not two. Then is the ring $$k[x,y,z]/(x^2+y^2-z^2)$$ a UFD? I admit that $k[x,y,z]/(xy-z^2)$ is not a UFD.
10
votes
1answer
351 views

Question about whether axiom of choice is needed in this proof

Do I need axiom of choice in this proof here? I think not: at each step we choose one element from a set $N - \langle g_1, \dots, g_k \rangle $. So while there is indeed a countable number of sets ...
10
votes
3answers
351 views

Is the coordinate ring of SL2 a UFD?

Is the ring $K[a,b,c,d]/(ad-bc-1)$ a unique factorization domain? I think this is a regular ring, so all of its localizations are UFDs by the Auslander–Buchsbaum theorem. However, I know there ...
10
votes
2answers
878 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
10
votes
1answer
923 views

Geometric meaning of primary decomposition

In the book "Commutative Algebra with a view toward Algebraic Geometry of David Eisenbud, he wrote about the Geometric interpretation of primary decomposition. I summary as follows : Let ...
10
votes
2answers
478 views

Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise ...
10
votes
1answer
1k views

Inverse limit of modules and tensor product

Let $(M_n)_n$ be an inverse system of finitely generated modules over a commutative ring $A$ and $I\subset A$ an ideal. When is the canonical homomorphism $$\left(\varprojlim\nolimits_n ...
10
votes
1answer
720 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
10
votes
1answer
540 views

A question on faithfully flat extension

This question arose while reading page 116 of Red Book by Mumford. Let $B$ be a faithfully flat extension of $A$. Can I claim that $b \otimes 1 = 1 \otimes b$ in $B\otimes_A B$ if and only if ...
10
votes
1answer
385 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb ...
10
votes
3answers
306 views

Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
10
votes
1answer
380 views

Finitely generated modules over PID

Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID $R$ such that $A\oplus $ $B$ $\cong$ $C\oplus $ $D$ and $A\oplus $ $D$ $\cong$ $C\oplus $ $B$ . Prove that $A$ $\cong$ $C$ and $B$ ...
10
votes
3answers
269 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
10
votes
2answers
488 views

What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?

Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find the primary decomposition of $I$. I tried to draw the graph of the variety of $I$ and get a decomposition of ...
10
votes
1answer
289 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...
10
votes
1answer
880 views

(Ir)reducibility criteria for homogeneous polynomials

Suppose I have a homogeneous polynomial in at least 3 variables over some algebraically closed field (of characteristic 0, if need be). Question: How may I test — by hand — whether it is irreducible? ...
10
votes
1answer
286 views

Intersection of powers of maximal ideals

Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid ...
10
votes
1answer
235 views

When is $\mathbb{Z}$ a flat $\mathbb{Z}G$-module?

Suppose that $\mathbb{Z}$ is a flat $\mathbb{Z}G$-module for a group $G$. Question: Is $G$ the trivial group ? Nb. I know that the question can be answered affirmatively if $G$ is finitely ...
10
votes
2answers
270 views

A commutative group structure on $R\times R$ for a ring $R$

Let $R$ be a commutative ring. The Cartesian square $A=R\times R$ is endowed with the operation $(a_1,b_1)\circ(a_2,b_2)=(a_1+a_2,b_1+b_2+a_1a_2^2+a_1^2a_2)$ which turns $A$ into a commutative ...
10
votes
1answer
373 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
10
votes
1answer
659 views

Geometric meaning of completion and localization

Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions. The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
10
votes
2answers
494 views

Intersection of finitely generated ideals

Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example? Thanks.
10
votes
2answers
314 views

Minimal systems of generators for finitely generated algebras over commutative (graded) rings

Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that ...
10
votes
3answers
327 views

How does one prove that the ring of integer-valued polynomials $\text{Int}(\mathbb{Z})$ is not Noetherian?

How does one prove that the ring of integer-valued polynomials $\text{Int}(\mathbb{Z})$ is not Noetherian? I let $(1, f_1, ..., f_n,...)$ be the $\mathbb{Z}$-basis of $\text{Int}(\mathbb{Z})$, ...
10
votes
2answers
136 views

Contents of Tor modules

I'm interested in knowing a concrete description of what elements of Tor modules $\mathrm{Tor}^i_R(M,N)$ "are". As it stands I have no real intuition for, say, maps between Tor modules induced by ...
10
votes
1answer
578 views

Homomorphism of local rings

Let $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ be local Noetherian rings. Suppose that $\phi : A\rightarrow B$ is a map such that $\phi(\mathfrak{m}) \subset \mathfrak{n}$ and suppose ...
10
votes
2answers
367 views

Computing intersection multiplicity using Tor - explicit example

When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$: Let $k$ be an algebrically ...
10
votes
1answer
269 views

Minimal spectrum of a commutative ring

Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
10
votes
1answer
43 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
10
votes
2answers
147 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
10
votes
1answer
249 views

Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?

I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the ...
10
votes
1answer
886 views

Intuition behind Hilbert's Nullstellensatz

maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot understand the concept in a more ...
10
votes
1answer
221 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
10
votes
2answers
1k views

When is a tensor product of two commutative rings noetherian?

In particular, I'm told if $k$ is commutative (ring), $R$ and $S$ are commutative $k$-algebras such that $R$ is noetherian, and $S$ is a finitely generated $k$-algebra, then the tensor product ...
10
votes
1answer
130 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...