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

learn more… | top users | synonyms (1)

2
votes
1answer
131 views

Construction of a Valuation Ring via Zorn's Lemma, except not

In Atiyah & MacDonald they provide an abstract way to construct valuation rings, I'm curious how easy it is to work this out in general. As a refresher let $K$ be a field, $L$ an algebraically ...
0
votes
2answers
89 views

Function field has infinitely many valuations.

Suppose we have a field $K$ which is a finite algebraic extension of the field $\mathbb{C}(X)$. Can you give me an argument that $K$ admits infinitly many discrete valuations?
2
votes
0answers
67 views

A injective Endomorphism over an Artinian Module is an Automorphism [duplicate]

Possible Duplicate: If $M$ is an artinian module and $f$ : $M$ $\mapsto$ $M$ is an injective homomorphism , then f is surjective Let $R$ be a commutative Ring with identity. We have an ...
4
votes
1answer
111 views

Existence of a root in $k[x_1, \ldots, x_n]$

Prove the following: If $k$ is an algebraically closed field and $f(x_1, \ldots, x_n) \in k[x_1,\ldots, x_n]$ is non-zero, then there exists $(a_1, \ldots, a_n)\in k^n$ s.t $f(a_1, \ldots, a_n) = 0$. ...
8
votes
1answer
773 views

Finitely generated algebra

I am getting the confusion with the definition of algebra. When we say $A$ is a finitely generated $R$- algebra then is that mean $A$ has a ring structure and finitely generated as an $R$-module. ...
3
votes
2answers
185 views

Krull dimension of the injective hull of residue field

Let $(R,\mathfrak{m})$ be a noetherian local ring, and $E=E_R(R/\mathfrak{m})$ the injective hull of $R/\mathfrak{m}$. What do we know about the Krull dimension of $E$? Thank you.
7
votes
1answer
194 views

The Picard-Brauer short exact sequence

It seems to be a rather well understood fact that, given commutative rings $R,S$, and a homomorphism $R \to S$ there is a short exact sequence $$\text{Pic}(R) \to \text{Pic}(S) \to F_0 \to ...
7
votes
2answers
277 views

Non-trivial valuation of $\mathbb R$

In a valued fields book on page $82$ there is a question: "show that every non-trivial valuation of $\mathbb R$ has divisible value group and algebraically closed residue class field." How do I ...
3
votes
0answers
104 views

A question about integral domains and inclusion relations in abstract algebra

Let $R$ be a Prüfer domain with quotient field $K$ and $\sum$ be the set of all semilocal Prüfer domains $R'$ with $n$ maximal ideals and quotient field $K$ such that $R\subseteq R'$. Let ...
5
votes
1answer
154 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
0
votes
0answers
52 views

The $I$-torsion submodules of an injective module [duplicate]

Possible Duplicate: Prove that the following module is injective Prove that $I$-torsion submodules of injective modules are injective (the ring is Noetherian). Please help me.
1
vote
2answers
137 views

A question about localization of integral domains

Let $R$ be an integral domain and $P,Q$ be proper prime ideals of $R$. Let $R_P,R_Q$ be localizations of $R$ at $P,Q$. If $R_P\subseteq R_Q$, is $Q\subseteq P$?
4
votes
1answer
170 views

Prove that the following module is injective

Let $R$ be a commutative Noetherian ring with identity. Prove that if $I$ is an ideal of $R$ and $E$ an injective $R$-module, then $\bigcup_{n\geq 1}(0:_{E}I^{n})$ is an injective $R$-module. Please ...
5
votes
1answer
290 views

What is an example of a radical of sum of ideals not being equal to the sum of radicals?

What is an example of the radical of a sum of ideals not equal to sum of the radical of the ideals?
4
votes
0answers
201 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
9
votes
1answer
205 views

Noetherian ring whose ideals have arbitrarily large number of generators

Does a commutative ring satisfying the following two properties exist? All ideals are finitely generated; There are prime ideals with arbitrarily large (finite) minimal generating sets.
3
votes
1answer
77 views

Real valuations on Dedekind domains

Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation. We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR. Does every valuation on $D$ ...
1
vote
0answers
104 views

Maximal ideals completion

I need help with a problem dealing with completion. It's hard for me to understand how to use the theory for specific examples, so help would be appreciated. Let $R = \mathbb{C}[x,y,z]/ (zy^2-x^3)$ ...
2
votes
1answer
102 views

Why, when considering prime ideals in the set $(\mathfrak a:x)$ $(x\in A)$ we may assume $\mathfrak a=0$?

Let $\mathfrak a$ be an ideal of a Noetherian ring $A$ and $x$ an element in $A$. Define $(\mathfrak a:x)=\{r\in A:rx\in \mathfrak a\}$. I try to understand one thing about the ideals in the set ...
3
votes
1answer
88 views

$I|J \iff I \supseteq J$ using localisation?

Let $R$ be a Dedekind domain. We know that for ideals $I$ and $J$ of $R$ that $I|J \iff I \supseteq J$. This fact is used for example in Marcus' Number Fields to show that we have unique factorisation ...
5
votes
1answer
72 views

A prime poset of ideals

Let $A$ be a ring (commutative unital), and $\mathcal I$ be a nonempty family of proper ideals of $A$. I will say that $\mathcal I$ has property $\dagger$ if for any $\mathfrak a\in\mathcal I$ and ...
1
vote
1answer
222 views

How to prove that this sub-algebra is not finitely generated?

Consider the subalgebra $R=k[x,xy,xy^2,xy^3, \ldots] \subset k[x,y]$. How do I prove that $R$ is not finitely generated (over $k$)? What is the general strategy for proving that an algebra is ...
3
votes
1answer
108 views

Special Case of Lying Over Theorem

In Pete Clark's commutative algebra lecture notes which can be found here. He proves the following lemma (14.12) Let $R$ be a local ring with maximal ideal $\mathfrak{p}$ and $S/R$ an integral ring ...
4
votes
1answer
120 views

$K[[x]]$ is not a Jacobson ring

Recall that a ring is called Jacobson if the radical of an ideal in the intersection of the maximal ideals that contains it (this is always true with prime ideals). $K[[x]]$ is not Jacobson. I ...
3
votes
1answer
168 views

Three kinds of spectra

In commutative algebra while proving the Nullstellensatz one introduces for a while the Rabinowitsch sprectrum as: $$\operatorname{Spec}_{\rm Rab}(R)=\{R\cap \mathfrak m: \mathfrak m \text{ is a ...
5
votes
2answers
286 views

Does finite projective dimension localize?

Let $R$ be a commutative (but not necessarily Noetherian) ring with unity. Let $M$ be an $R$-module. Suppose that, for all $\mathfrak p \in\text {Spec}(R),$ $\text{pd}_{R_{\mathfrak p}}M_{\mathfrak ...
9
votes
1answer
317 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 ...
7
votes
2answers
477 views

How to prove that this subring is not noetherian?

Consider the subring $R=k[x,xy,xy^2,\ldots]$ of $k[x,y]$. I want to prove that $R$ is not noetherian. An ascending chain of ideals is the following ...
4
votes
1answer
110 views

Henselization and immediate extension

I am reading about the henselization and immediate extension of valuation. I am getting confusion about some basic terminology. I have few question. \ $1)$ Is every hensilization extension of ...
0
votes
2answers
79 views

Proving $\operatorname{Spec} k[x_1,\cdots, x_r] \subset \operatorname{Spec} k[x_1,\cdots, x_{r+1}]$

Let $k$ be a field. Is there an elegant proof of the fact that $\operatorname{Spec} k[x_1,\cdots, x_r] \subset \operatorname{Spec} k[x_1,\cdots, x_{r+1}]$? I proved it as follows: let $P \in ...
0
votes
1answer
94 views

A zero-dimensional ring is Noetherian?

Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian. Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq ...
1
vote
1answer
170 views

Prove that if $M$ is an $R$-module of finite length, then $\operatorname{End}_R(M)$ is artinian

Prove that if $M$ is an $R$-module of finite length, then $\operatorname{End}_R(M)$ is artinian. We can derive that $M$ is both artinian and noetherian from that it has finite length, and its ...
3
votes
1answer
98 views

Integral extension (Exercise 4.9, M. Reid, Undergraduate Commutative Algebra)

Let $k$ be any field and let $A = k[X,Y,Z]/(X^2 - Y^3 - 1, XZ - 1)$. How can I find $\alpha, \beta \in k$ such that $A$ is integral over $B = k[X + \alpha Y + \beta Z]$? For these values of ...
2
votes
2answers
209 views

Modules over local ring and completion

I'm stuck again at a commutative algebra question. Would love some help with this completion business... We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the ...
2
votes
1answer
605 views

Finitely generated modules over artinian rings have finite length

Suppose $M$ is an $R$-module. Prove that $M$ has finite length if $R$ is artinian and $M$ is finitely generated.
7
votes
3answers
228 views

When does locally irreducible imply irreducible?

The situation is this: I have a homogeneous ideal with many generators and variables, too many to simply ask isPrime I in Macaulay2. However, the ideal simplifies ...
6
votes
2answers
330 views

Ring of formal power series finitely generated as algebra?

I'm asked if the ring of formal power series is finitely generated as a $K$-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
2
votes
1answer
44 views

Question on Integral Closure

I'm trying to prove this fact: given $A$ an integral domain and an element $f\in A$ such that $A/fA$ has no nilpotents, then $A$ is integrally closed if and only if $A_f$ is integrally closed ...
2
votes
1answer
100 views

Valuation over the algebraically closed field of rational number

How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
0
votes
1answer
39 views

If ideals $Q_1,Q_2$ lie over a prime in $\Bbb{Z}$ their product lies over the prime squared?

Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ...
1
vote
2answers
100 views

Adjoining an inverse to a local UFD of Krull dimension 2 gives a PID

Let $R$ be a local UFD of Krull dimension 2. Let $a\in R$ be a nonzero, non-unit. I am trying to show that the ring $R[1/a]$ is a principal ideal domain. Does anyone have any suggestions as to how ...
4
votes
1answer
161 views

Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism? I believe the answer should be yes, ...
3
votes
2answers
139 views

When can a ring imbed in a localization?

In Topics in Algebra, there is an exercise (3.6.5): let $R$ be a commutative, unital ring and let $S\subset R$ be non-empty and such that $s_1 s_2\in S$ if $s_1,s_2\in S$ and $0\not\in S$. Construct ...
1
vote
1answer
301 views

$(M\otimes_A N)_B \cong M_B\otimes_B N_B$?

Let $A \rightarrow B$ be a homomorphism of commutative rings. Let $M, N$ be $A$-modules. We denote $M\otimes_A B$ by $M_B$. We regard $M_B$ as a $B$-module. Then $(M\otimes_A N)_B \cong M_B\otimes_B ...
1
vote
2answers
80 views

a commutative ring statisfying an integer polynomial

Let $p(X)\in\mathbb{Z}[X]$ be a monic polynomial and let $A$ be a commutative ring in which every element is a zero of $p(X)$. Prove that all prime ideals in $A$ are maximal. By definition, give ...
5
votes
5answers
412 views

Is there a finitely generated, algebraic $K$-algebra $A$ that is not a field?

There is a well-known theorem that states that if $A$ is a finitely generated $K$-algebra, an integral domain and algebraic over $K$, then $A$ is a field. Is the integral domain condition necesary? I ...
2
votes
2answers
262 views

$\mathbb Z/n\mathbb Z$ is not a projective module

I want to show that $\mathbb Z/n\mathbb Z$ is not projective for $n\geq 2$. I choose the exact sequence $\mathbb Z\stackrel{\pi}\rightarrow\mathbb Z/n\mathbb Z\rightarrow 0,$ and from $\mathbb ...
5
votes
5answers
172 views

A question on local rings

I was trying to get a counterexample of this fact: given a ring $A$, $f\in A$ and $S=\{1,f,f^2,...\}$, is $S^{-1}A$ always a local ring? Could you help me please? Thank you.
2
votes
2answers
135 views

Irreducibility and Subspace Topology

Is the following statement true? "Let $X$ be a topological space, $Y$ a subspace and $S$ a closed and irreducible subset of $X$. Then $Y \cap S$ is not necessarily irreducible in $Y$." Counterexamples ...
2
votes
2answers
82 views

Can a ring of integers contain a $2$-dimensional noetherian normal integral domain?

Let $K$ be a number field with ring of integers $O_K$. Does there exist a $2$-dimensional subring $A\subset O_K$? Clearly, if such a subring $A\subset O_K$ exists, we have that $A$ is an integral ...