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

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3
votes
1answer
243 views

Ring of Fractions

Let $A$ be an integral domain. If $S=A-\left\{0\right\}$, then $S^{-1}A$ is the field of fractions of $A$. What is the problem if we actually take $S=A$? From what i see, in that case $0/0=1/1$ and ...
1
vote
1answer
350 views

Tensor product of a finitely generated modul and a finite length module is finite length

Let $R$ be a commutative ring and $M,N$ $R$-modules finitely generated with $M$ of finite length. How can I prove that $M\otimes_R N$ is of finite length?
2
votes
1answer
247 views

Turning the tensor product of algebras into an algebra

Let $B, C$ be $A$-algebras, where $A$ is a commutative ring, i.e. $B, C$ are rings and we have ring homomorphisms $f:A\rightarrow B, g:A \rightarrow C$. Since both $B, C$ are $A$-modules, we define ...
4
votes
2answers
336 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
1
vote
1answer
80 views

Understanding a morphism of modules by properties of the induced residue field homomorphism

Let $A$ be a reduced local Noetherian ring, and $\phi: M\to N$ a morphism of finitely generated free $A$-modules. For all $\mathfrak{p}\in\text{Spec}(A)$, let ...
1
vote
1answer
226 views

indecomposable module which is not cyclic

In Etingof's notes entitled "Introduction to Representation Theory," he asks the reader to produce an example of an indecomposable module which is not cyclic (Problem 1.25(c)). The exercise even comes ...
3
votes
0answers
405 views

An algebra of finite type over a field is a Jacobson ring.

Let $R$ be an algebra of finite type over a field $k$. I want to show that $R$ is Jacobson, i.e. that any prime ideal $\mathfrak{p}$ in $R$ is an intersection of maximal ideals. I am not getting very ...
10
votes
3answers
1k views

About the localization of a UFD

I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, what about ...
1
vote
2answers
192 views

$(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$

Let $B, C, D, E$ be $A$-modules. Is there a way to show that $(B \otimes C) \otimes (D \otimes E)$ is isomorphic to $B \otimes C \otimes D \otimes E$ using the result that $(M \otimes N) \otimes P$ ...
3
votes
1answer
219 views

Not a radical ideal

Is there a method to compute the radical of an ideal? for example take $J=(xw-y^{2},xw^{2}-z^{3}) \subset k[x,y,z,w]$. I want to show $J$ is not radical, I guess the idea is to add and substract terms ...
8
votes
0answers
529 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
5
votes
1answer
184 views

Tensor-free proof that for finite modules over reduced Noetherian rings, locally free = projective

Is there an elegant tensor-free proof of the fact that over a reduced Noetherian ring $A$, every finitely-generated $A$-module which is locally free, is projective? EDIT: I would be content with the ...
0
votes
3answers
1k views

Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.

Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain. We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
8
votes
4answers
798 views

What is the fraction field of $R[[x]]$, the power series over some integral domain?

I have a question similar to 74335. Let $R$ be an integral domain. Is there a nice description of the fraction field of the power series $R[[x]]$? I know that this field can be a proper subfield ...
1
vote
0answers
93 views

subrings A of the ring of power series k[[t]] with the condition (A : k[[t]]) $\neq${0} and k $\subset$ A

I would like to understand the structure of the subrings A of the ring of formal power series k[[t]] (where k is a field) which satisfy the condition (A : k[[t]]) $\neq$ {0} and k $\subset$ A. Are ...
2
votes
1answer
234 views

Integral domain and ascending chain condition proof

Show that an integral domain $A$ is a principal ideal domain if every ideal $I$ of $A$ is principal, that is, of the form $I=(a)$. Show directly that the ideals in a PID satisfy the a.c.c.
5
votes
3answers
164 views

General Form of $S^{-1}A$ - modules

I have been trying to show that if a ring $A$ is absolutely flat then so is the localisation $S^{-1}A$ by any multiplicative set. Now while trying to do this, I asked myself the following: Is there a ...
7
votes
2answers
293 views

Is this quotient ring $\mathbb{C}[z_{ij}]/\ker\phi$ integrally closed?

A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm ...
6
votes
0answers
126 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
2
votes
2answers
192 views

What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)

Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
2
votes
2answers
371 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
2
votes
0answers
67 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
2
votes
0answers
64 views

the algebra of “p-th roots”

Let $R$ be an integral algebra of finite type over a (perfect) field of characteristic $p > 0$. Let $S$ be the integral closure of $R$ in $Q(R)^{1/p}$ where $Q(R)$ is the field of fractions of $R$. ...
5
votes
1answer
79 views

How to extract roots in a complete local ring using binomial series

Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is ...
4
votes
0answers
67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
2
votes
1answer
69 views

Under some conditions, $K$ is algebraically closed in $K(x, y)$

Let $K$ be a field and $L = K(x, y)$, where $x$ is transcendental over $K$ and $y$ is such that $f(x, y) = 0$, for $f \in K[X, Y]$ irreducible. I have to prove that if $f$ is also irreducible over ...
11
votes
2answers
319 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 ...
4
votes
2answers
177 views

Does totally flat commutative ring imply all ideals are idempotent?

From reading Atiyah and MacDonald, I know of the result that a absolutely flat commutative ring has all principal ideals idempotent. Reading around on math reference, I think that if a commutative ...
15
votes
2answers
1k views

A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
5
votes
1answer
254 views

Finding the radical of some ideals

I need to find the radicals of the following ideals: i) $\mathfrak{a} = (xy^3, x(x-y))$ ii) $\mathfrak{b} = (xy^3, x^2(y-3))$ iii) $\mathfrak{c} = (x^2(y-z), xy(y-z), xz(y-z)^2)$ Can I just use ...
5
votes
1answer
218 views

Notation in Atiyah - Macdonald

I am now going through some problems in Atiyah - Macdonald Chapter 3. In problems 21 and 23 of chapter 3, they use the notation $A_f$ to mean something I don't know. I have not seen this before. ...
10
votes
0answers
157 views

checking that an element of a module is zero, point-wise

Let $M$ be a module over a commutative ring $R$. Let $s \in M$ be an element such that for any $x \in \mathrm{Spec}\,R$, the image of $s$ in $M \otimes \kappa(x)$ is 0 (where $\kappa(x)$ is the ...
6
votes
2answers
209 views

An $R$ module and $S$ module that cannot be an $R$-$S$ bimodule

In connection with this question: Modules and tensor products Question: For two commutative rings $R$ and $S$ (with unity), is there an abelian group $M$ which has $R$ module and $S$ module ...
2
votes
3answers
207 views

When is a module over $R$ and $S$ an $R \otimes S$-module?

Suppose $M$ is a module over $R$ and $S$, commutative rings with $1$. Under what conditions is $M$ also a $R \otimes S$-module? Also, a more general question: How to construct a structure of a $R ...
9
votes
1answer
479 views

How to tell if an element of a quotient ring is a zero divisor

I am looking at Hartshorne Example III.9.8.4., p260. He says that $a$ is not a zero divisor in $k[a,x,y,z]/I$, where $$ I = (a^2(x+1) -z^2, ax(x+1)-yz, xz-ay,y^2-x^2(x+1)). $$ Is there a good way to ...
3
votes
1answer
99 views

How to show that $M_B = B \otimes_{A} M$ is a $B$-module?

Let $A,B$ be commutative rings with identity. Let $f:A \rightarrow B$ be a ring homomorphism and let $M$ be an $A$-module. Since $B$ can be viewed as an $A$-module with the operation $A \times B ...
7
votes
3answers
370 views

Module M/IM of finite length $\implies$ Ring A/I of finite length

This question is due to a proof in an algebra book (on the topic of dimension theory) which I don't fully understand (specifically, the proof of Thm 6.9b) in Kommutative Algebra by Ischebeck). It may ...
10
votes
1answer
638 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 ...
2
votes
2answers
427 views

Isomorphism of tensor product+field extension

Let $k$ be a field, $f(x)\in k[x]$ be an irreducible polynomial over $k$, and $\alpha$ be a root of $f$. If $L$ is a field extension to $k$, what does $k(\alpha)\otimes_k L$ isomorphic to? I'm ...
5
votes
1answer
105 views

Question about regular local rings

Let $A$ be a commutative regular local ring of dimension $d$ with maximal ideal $\mathcal m$ and $a \in A$ an element of the ring. Suppose that $\mathcal m \cdot a \subset \mathcal m^2$, i.e. if I ...
0
votes
1answer
170 views

homomorphism of Laurent polynomial ring

This question is similar to the question link. Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Given $r\in \mathbb Z_+$ and $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$, ...
6
votes
2answers
280 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
1
vote
1answer
245 views

Standard graded algebra

I am so sorry if you feel this kind of question is not appropriate for MS. But I hope you can sympathize with me, I tried to find the answer in all my books and even Google but I found nothing. My ...
2
votes
2answers
189 views

About Gorenstein ring

Is it true that in a (non-local) Gorenstein ring, every maximal ideal has the same height? It seems a little strange, but I don't see any reason why it shoudn't.
5
votes
2answers
751 views

Why this element in this tensor product is not zero?

$R=k[[x,y]]/(xy)$, $k$ a field. This ring is local with maximal ideal $m=(x,y)R$. Then the book proves that $x\otimes y\in m\otimes m$ is not zero, but I don't understand what's going on, if the ...
2
votes
2answers
353 views

Every endomorphism of a finitely generated module satisfies a polynomial equation.

I encountered the following very interesting proposition in Atiyah's and McDonald's Introduction to Commutative Algebra: Let $A$ be commutative ring with identity, $M$ a finitely generated ...
6
votes
3answers
592 views

Alternative construction of Direct Limit

The construction of the direct limit that I learned from Atiyah Macdonald is the following: Suppose we have a directed system $(M_i,\mu_{ij})$ of $A$ - modules and $A$ - module homomorphims over a ...
6
votes
1answer
1k views

A free submodule of a free module having greater rank the submodule

Let $R$ be a commutative ring, and let $N\leq M$ be $R$-modules. Then, suppose $M$ and $N$ are free over $R$, if $R$ is an integral domain, then -considering the fraction modules over the quotient ...
2
votes
1answer
1k views

Commutative Ring: Nilpotent elements closed under addition? [duplicate]

Possible Duplicate: The set of all nilpotent element is an ideal of R Given a commutative ring $R$ and two nilpotent elements $r$, $s$ there exists an $n \in \mathbb{N}$ such that $$ ...
2
votes
0answers
268 views

Injective module and Noetherian ring

In the book Abstract Algebra of J.Antoine Grillet there is a theorem as follow: A ring R is left Noetherian if and only if every direct sum of injective left R-modules is injective The ...