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

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9
votes
1answer
691 views

Fraction field of the formal power series ring in finitely many variables

What is the fraction field of the formal power series ring over a field in finitely many variables $K[[X_1,\dots,X_n]]$? Is there a nice description for this field? When $n=1$, I know this is the ...
1
vote
2answers
188 views

Localisation contained in completion?

I'm working on an exercise in which I have to show that localising and completing are exact functors. More precisely I have a Dedekind domain $R$ and a prime ideal $\mathfrak{p}$ and I have to show ...
0
votes
1answer
83 views

On unique factorizations of ideals

Using standard notations, let $K$ be a number field and $S = \left\{p_{1}, ..., p_{n}\right\}$ a finite set of non-zero prime ideals of $K$. Let $a$ be a non-zero fractional ideal of $K$. Prove that ...
4
votes
2answers
292 views

Different version of Gauss's Lemma

Let $A$ be a domain with field of fractions $K$. Let $f, g \in A[X]$ with $g$ monic. Show that if $f/g \in K[X]$ then $f/g \in A[X]$. So I tried the direct approach by just assuming $f/g$ has a ...
1
vote
1answer
79 views

$(a_1,\ldots,a_n)\!=\!(a)\;\Leftrightarrow\;a\!=\!\gcd(a_1,\ldots,a_n)$?

Could you please help me finish the proof below. The only problem is the $(\Leftarrow)$ part of a). Proposition???: In any domain: a) $(a_1,\ldots,a_n)\!=\!(a)\;\Leftrightarrow\;a\!=\!\gcd(a_1,\...
4
votes
1answer
961 views

Left exactness of inverse limit

Is the left exactness of inverse limit (in the category of modules over a ring) a general property regardless of the indexing set? (Let's assume it is still directed.) The only proof I can find ...
3
votes
3answers
368 views

A Noetherian Ring with Discrete Spectrum is Artinian

I'm trying to solve an exercise. I should prove that if $R$ is a notherian ring and $\operatorname{Spec}(R)$ is discrete then $R$ is artinian. I think it is enough to show that $\dim R=0$ ...
2
votes
1answer
222 views

Quotient Rings and Integral Extensions

Suppose $S$ is an integral extension of $R$ and $I$ an ideal in $S$. Why is $S/I$ an integral ring extension of $R/(R \cap I)$? To this question, Dummit and Foote says: Reducing the monic ...
0
votes
1answer
66 views

intersections of powers of primes lying over a prime in a Galois extension

Suppose $A$ is a Dedekind domain with fraction field $K$ and $L/K$ is Galois, let $B$ be the integral closure of $A$ in $L$. Let $P$ be a prime ideal in $A$ and let $P_1,...,P_n$ be prime ideals ...
13
votes
2answers
1k views

A non-noetherian ring with all localizations noetherian

If for a ring $A$ every localization $A_\mathfrak{p}$ by a prime $\mathfrak{p}\subseteq A$ is noetherian, is it true that $A$ is noetherian? I believe not but I can't find a good counterexample.
4
votes
1answer
85 views

algebra homomorphism $k^S \to k$

Let $k$ be a field and $S$ be an infinite set. Assume $|S| \leq |k|$. Why is then every $k$-algebra homomorphism $k^S \to k$ equal to a projection $\mathrm{pr}_s$ for some $s \in S$? I don't know how ...
4
votes
1answer
132 views

Containment of primary ideals

Suppose, $R$ is a noetherian ring. Let $P$ be a prime ideal in $R$. Let $Q$ be a $P$-primary ideal that contains $P^n$. Then does $Q$ contain $P^{(n)}$ which is the $n$th symbolic power of $P$ and is ...
3
votes
1answer
253 views

Existence of inverse limit in an arbitrary category

According to Wikipedia article http://en.wikipedia.org/wiki/Inverse_limit "Unlike for algebraic objects, the inverse limit might not exist in an arbitrary category." But when constructing the ...
6
votes
2answers
186 views

Associated primes of a sum of modules

Let $M$ be a module with $M_1$ and $M_2$ submodules such that their sum (not necessarily a direct sum) is $M$. Is it true in full generality that $\text{Ass}(M) = \text{Ass}(M_1) \cup \text{Ass}(M_2)$?...
1
vote
1answer
70 views

Radical of an ideal after adjoining roots

Let $A$ be a Noetherian domain containing an algebraically closed field $k$. Let $x_1,\ldots,x_r\in A$ be irreducible elements generating a radical ideal $I=(x_1,\ldots,x_r)$. Set $B:=A[y_1,\ldots,y_r]...
3
votes
2answers
473 views

Projective Spectrum of $K[X,Y]$

Let's assume that $K$ is algebraically closed. I'm having some difficulties figuring out what $\text{proj}\;K[X,Y]$ is, where $K[X,Y]$ is interpreted as a graded ring. Any hints? So far I have only ...
3
votes
2answers
2k views

Ideal generated by a irreducible element

Is the ideal generated by an irreducible element always a prime ideal in a ring? If so why?
6
votes
2answers
1k views

If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.

In this question I will use the following definition of a Dedekind domain: An integral domain $A$ is a Dedekind Domain if: 1) $A$ is a Noetherian Ring. 2) $A$ is integrally closed. 3) Every non-...
2
votes
1answer
161 views

About injective hull of residue field

Let $(A,\mathfrak{m})$ be a noetherian local ring, and $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. I'm pretty sure that $E(A/\mathfrak{m})$ doesn't automatically extend to an $A/\...
10
votes
1answer
1k views

A question about the tensor product of $\mathbb{Q}$

I'm reading this blog post about $\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q}$ and I have two questions about it: Is a simple tensor a tensor that cannot be written as a sum of tensors? On the first ...
9
votes
2answers
532 views

Motivation behind the definition of localization

What is the motivation behind definition of localization of rings? From where does the term "localization" come from? Why is the equivalence relation between the ordered pairs $(m,u),(m',u')$ with $ ...
1
vote
1answer
321 views

epimorphism in the category of commutative rings

Let $\phi:A\to B$ be an epimorphism in the category of commutative rings, we can find that the induced continuous map $\phi^*$ from Spec$B$ to Spec$A$ is injective as a map between sets, I want ...
1
vote
1answer
161 views

Particular case of Nakayama's lemma

I'm trying to prove the following particular case of Nakayama's lemma. Let $R$ be a commutative ring and $a\in R$ be nilpotent (let's suppose $a^{k-1}\not=0$, $a^k=0$). Then $aM=M \Rightarrow M=\{0\}$....
15
votes
2answers
841 views

Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
3
votes
1answer
640 views

Why does surjectivity of the induced map show that a morphism of affine varieties has closed image?

Let $\phi : X \rightarrow Y$ be a morphism of affine varieties and let $\phi^\ast : k[Y] \rightarrow k[X]$ be the induced map on coordinate rings. My text says that if $\phi^\ast$ is surjective then $...
1
vote
1answer
137 views

Question about Going-Up

Let $k$ be a field, not necessarily algebraically closed. Then how would you show that the extension $k[x] \subset k[x,y]$ does or does not satisfy Going-Up?
1
vote
1answer
50 views

Uniqueness of representation of an element of an ideal in domains

Suppose $R$ is a domain and $I=aR$ be a non-zero principal ideal. Then, every element of $I$ has a unique representation, for if $ra=sa$ then $(r-s)a=0$. Since, $a\neq 0$ and $R$ is a domain, we have, ...
4
votes
0answers
185 views

Prime ideals in galois extensions

This is with reference to proposition 1 in Robert Ash's notes I don't think the Dedekind assumption is necessary. Explicitly, if $A$ is an integral domain with fraction field $K$ and $L/K$ is galois, ...
2
votes
0answers
78 views

If $P$ is a prime ideal in a ring, then its only associated prime is the ideal itself?

This seems obvious to me, but I don't see it explicitly mentioned, so I may be wrong here, but since a prime ideal is primary, we have a primary decomposition for a prime ideal consisting of just the ...
3
votes
1answer
174 views

Am I wrong in thinking it is isomorphism rather than homomorphism?

The following is a quotation from the proof of Proposition 11.10 in "Introduction to Commutative Algebra" by Atiyah and MacDonald. Also if ${\mathfrak m}'$ is the maximal ideal of $A'$, $A'/...
5
votes
0answers
756 views

What is a linear resolution?

Can anyone tell me where I may find an introduction to linear resolutions (of a $k[x_1,\ldots,x_n]$-module or ideal) including, of course, the standard definition of such a resolution, as well as its ...
16
votes
3answers
646 views

*writing* proofs involving commutative diagrams

This question is a little fuzzy so might be closed, but I'll give it a shot. I'm sorry this question has quite a long introduction, I don't see how to formulate it more concisely. In modern algebraic ...
6
votes
1answer
117 views

Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one: Are there any integrally closed BFDs which are not Krull domains? Some background information: A BFD (bounded factorization domain) is ...
5
votes
2answers
520 views

Example of non-Noetherian non-UFD Krull domain?

After a confusing session of hopping through Wikipedia articles, I started trying to summarize for myself some of the inclusions and relations among the many types of integral domains. Right now I'm ...
6
votes
3answers
262 views

Involutions on commutative rings

I found that all the commutative rings with involution I know are the following: complex number with complex conjugation (plus similar constructions based on rationals and its extensions), any ...
2
votes
1answer
127 views

Going-up property

Given commutative rings $A$ and $B$, if $B$ is an $A$-algebra, under what conditions, other than $B$ being integral over $A$, will the Going-Up property hold? Is there a condition weaker than ...
8
votes
4answers
1k views

Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
5
votes
1answer
475 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 $W \subset V \cap U$ ...
0
votes
1answer
194 views

Determining the minimal number of generators of the maximal ideal of a local Noetherian ring

Let ($A,m$) be a local, Noetherian ring. If $n$ is the minimal number of generators of the unique maximal ideal $m$, then by Krull's Hauptidealsatz and Nakayama's Lemma, we have the following ...
2
votes
1answer
102 views

When is an affine algebra normal?

I was looking at results that give necessary (and possibly sufficient) conditions on an ideal $I$ in the ring $R=k[x_1,x_2,...,x_n]$ such that $R/I$ is normal. I don't know if there is a standard ...
0
votes
1answer
52 views

If $Rm$ is free and $N$ is free is $ m \otimes n \neq 0$?

This post is a follow up to the counterexample presented in the following questions If $Rm$ is free, how do you show $m \otimes n \neq 0$?. The hope now is that we can eliminate the pathological ...
4
votes
2answers
334 views

Showing $R(m\otimes n)$ is free if $Rm$ and $Rn$ are free

Let $R$ be a commutative ring with identity. The following is a statement I came across about the submodule $Rt$ generated by a decomposable tensor $t=m\otimes n$ being free, given that $Rm$ and $Rn$...
3
votes
1answer
90 views

Generators for $M_n(\mathbb Q)$

What is the minimum number of generators for $M_n(\mathbb Q)$, the set of $n \times n$ matrices over $\mathbb Q$, which will generate it as an algebra over $\mathbb Q$ ?
8
votes
1answer
130 views

Injective map between power series ring

Suppose $k$ is a field and let $n > m$. Does there exist injective homomorphisms $$ k [[x_1, x_2, \ldots, x_n]] \rightarrow k[[x_1, x_2, \ldots, x_m]]\ ?$$
6
votes
1answer
318 views

Does maximal Cohen-Macaulay modules localize?

Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $$\operatorname{depth}M= \dim M=\dim A.$$ I can prove that $$\operatorname{depth}M_{\mathfrak{p}}= \dim M_{\...
2
votes
4answers
134 views

The valuation ring $R$ in $K(T)$, such that $K[T] \subsetneq R \subsetneq K(T)$

$K$ is an algebraically closed field, $K[T]$ is the ring of polynomials of one indeterminate over $K$, and $K(T)$ is its field of fractions. A valuation ring $R$ in $K(T)$ which includes $k[T]$ and is ...
18
votes
5answers
7k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
5
votes
1answer
564 views

A wrong proof about Dedekind domains

I "proved" that a Dedekind domain is a PID, but as we know this is wrong (for example $\mathbb{Z}[\sqrt{-5}]$). I do not know what is wrong in my proof: Suppose $R$ is a Dedekind domain, $I$ is ...
4
votes
1answer
144 views

the reduced locus of a Noetherian ring

Let A be a Noetherian ring, Is the set of prime ideals $\{p\in \operatorname{Spec} A| A_p$ is a reduced ring $\}$ an open subset of $\operatorname{Spec} A$ in Zariski topology?
5
votes
1answer
140 views

Some question ideal of variety

For an affine variety $X=V(x^{2}+y^{2}-1, x-1)$, I found the ideal of $X$, $I(X)=\langle x-1,y\rangle$. But I don't know $I(X)=\langle x^{2}+y^{2}-1, x-1\rangle$.