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

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2
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3answers
2k 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
1k 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
106 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
255 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
167 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
346 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 ...
13
votes
0answers
157 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 ...
3
votes
2answers
220 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
604 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
72 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 ...
5
votes
1answer
84 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 $\...
12
votes
2answers
364 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 ...
6
votes
2answers
249 views

Does absolutely 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 ...
18
votes
2answers
2k 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 hints?
6
votes
1answer
384 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
235 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. ...
14
votes
0answers
170 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
240 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
224 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 \...
10
votes
1answer
633 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
111 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 \...
8
votes
3answers
435 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
690 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 $x+...
3
votes
2answers
585 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
124 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
184 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
305 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
334 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 ...
3
votes
2answers
219 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.
6
votes
2answers
944 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 ...
3
votes
2answers
461 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 $A$-...
8
votes
3answers
707 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 ...
7
votes
1answer
2k 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
2k 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 $$ (r+s)^n =...
5
votes
1answer
351 views

Injective module and Noetherian ring

In the book Abstract Algebra of J.Antoine Grillet there is a theorem as follows: A ring $R$ is left Noetherian if and only if every direct sum of injective left R-modules is injective The ...
1
vote
1answer
49 views

Doubt about completeness

May I refer you to page 3 of: http://www.math.iitb.ac.in/atm/caag1/balwant.pdf Proof that $\hat{M}$ is complete, where it says "We choose $n(m)$ such that $n(m+1) \geq n(m)$ for every $m$". ...
2
votes
2answers
1k views

nilpotent ideals [duplicate]

Possible Duplicate: The set of all nilpotent element is an ideal of R An element $a$ of a ring $R$ is nilpotent if $a^n = 0$ for some positive integer $n$. Let $R$ be a commutative ring, and ...
8
votes
2answers
607 views

Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective ...
6
votes
1answer
276 views

Automorphisms of $k[x_1,x_2,\dots,x_n]$ that fix $k$

Given a field $k$, consider the polynomial ring $k[x_1,x_2,\dots,x_n]$. Is it possible to find all the automorphisms of this ring that fix the field $k$?
0
votes
2answers
101 views

Problem on multiplicative subset

Let $R$ be a ring, $S$ is a multiplicative subset of $R$. $a$ is an arbitrary element of $S$. Should there be 2 element $b,c \in S, b, c \neq 1$ such that $a=b.c$? If not please give a counter ...
2
votes
1answer
178 views

Lemma on finite generation of algebras over a field

I saw this lemma in some lecture notes, there was no proof given nor a reference, only a statement that it can be found in any text-book on commutative algebra. I checked several but couldn't find it. ...
4
votes
1answer
159 views

Algebraic maps from products of affine varieties

I have a question which might be fairly elementary, but I could not find an answer in the literature yet. Any pointers are very welcome :) Let $X$, $Y$ and $Z$ be affine algebraic varieties. I have a ...
4
votes
2answers
347 views

Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
3
votes
1answer
199 views

On Krull dimension of $M/(0 :_{M} \mathfrak{m}^t)$ module

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module. There is an non-negative integer $t$ such that $M/(0 :_{M} \mathfrak{m}^t)$ is finitely generated. Then $$\dim ...
9
votes
2answers
251 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
4
votes
2answers
169 views

Infinite many curves passing through finite points?

Let $R$ be a Noetherian domain of dimension two. Let $\mathfrak{m}_1,\mathfrak{m}_2$ be two disctinct maximal ideals of height two. Are there always infinitely many prime ideals contained in $\...
3
votes
0answers
189 views

Is an irreducible element still irreducible under localization?

Suppose $R$ is a domain. We say an element $x\in R$ is "irreducible" if $x=yz$ implies that $y$ or $z$ is a unit or both are units. I want to know if an irreducible element is still an irreducible ...
1
vote
1answer
143 views

How to find the generic initial ideal?

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra, pp. 26-27. Let $f,g\in k[x_1,x_2,x_3,x_4]$ be generic forms of degree $d$ and $e$. The generic initial ideal of $I=\...