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

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1answer
106 views

Ideals of $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$

I was wondering if the ring $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$ is a PID. I believe that it is because I think $\mathbb{Q}(\sqrt[3]{2}) ...
2
votes
1answer
90 views

Powers of ideals in a polynomial ring

Let $F$ be a field, and consider the polynomial ring $F[X_1,...,X_n]$. I am trying to prove that every power of the ideal $(X_1,...,X_k)$ is primary for $k\leq n$. For $k = n$, we have that ...
3
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0answers
55 views

Proof about affine varieties

Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
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0answers
51 views

Rings noetherianos

Let $K$ be a field. Show that any subring of $K[X_{1},...,X_{n}]$ that it contains to $K$ is noetheriano. It gives an example in the one that is demonstrated not all these subrings are DFU.
4
votes
3answers
686 views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
5
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2answers
212 views

Is this a prime Ideal?

I wish to see wether $J=(uw -v^2, u^3 - vw, w^3 -u^5)\subset\mathbb{C}[u,v,w]$ is a prime ideal. Can somebody give me a hint to do this? Edit: More generally, I wonder wether $V(J)$, the algebraic ...
0
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0answers
37 views

$\mathtt{maxSpec}(\mathbb{Z}[x])=\{(p,g)\mid p\text{ is prime, }g\text{ mod }p \text{ is irred.}\}$ [duplicate]

I'm trying to prove this. This is my approach. Since $\mathbb{F}_p[x]/(\bar{g}) \cong \mathbb{Z}[x]/(p,g)$, $\mathtt{maxSpec}(\mathbb{Z}[x])\supseteq\{(p,g)\mid p\text{ is prime, }g\text{ mod }p ...
4
votes
2answers
90 views

Intersection of two localizations

Let $A$ be a commutative ring with unity. If $\mathfrak p,\mathfrak q\in \operatorname{Spec} (A)$ is it true the following equality $$A_\mathfrak p\cap A_\mathfrak q= A_{\mathfrak p\cup \mathfrak ...
4
votes
1answer
166 views

Noether Normalization Lemma for affine scheme over DVR?

Let $R$ be a DVR and $S$ a finitely generated flat $R$-algebra. How can I prove that there is a subalgebra $C$ of $S$ such that there is a finite and injective morphism $R[t_1,\dots, t_d] \rightarrow ...
2
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1answer
58 views

Why is the spectrum of $\mathbb{C}[X,X^{-1}]$ equal to $\mathbb{C}^*$?

Can someone help me see why the following is true?: $$\operatorname{Spec}( \mathbb{C}[X,X^{-1}])= \mathbb{C}^*$$ It was stated in something I read but I don't know why it is true. Thanks for your ...
3
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1answer
112 views

Finding a toric variety of a cone

I'm trying to find the toric variety associated to the cone $\sigma_0$ which is the region in the real plane with $x\geq 0$ and $y-x\geq 0.$ I found that it's dual cone is $\check{\sigma_0}$ the ...
2
votes
2answers
50 views

limits of sequences of topological rings

Let $A$ be a ring and $I$ an ideal of $A$ such that $A$ is complete in the $I$-adic topology. Let $a \in I$. Then the sequence $y_n=1-a+a^2-a^3+\cdots+(-1)^n a^n$ converges in $A$. By definition of ...
5
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1answer
99 views

$P/P^2$ isomorphic to $R/P$ as $R$-modules

Let $P$ be an ideal of a ring $R$. When is it true that $P^n/P^{n+1}$ are isomorphic to $R/P$ as $R$-modules for any $n$? I was trying to show that for Dedekind domains the norm of ideals is a ...
6
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1answer
212 views

faithfully flat ring extensions where primes extend to primes

I am interested in unital ring homomorphisms (and classes thereof) $R \rightarrow S$ of commutative rings that have the following pair of properties: $S$ is faithfully flat as an $R$-module, and ...
0
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2answers
64 views

How to prove $x$ doesn't lie in $R_M$

Let $R$ be an integral domain. $K$ is the field of fractions of $R$. Let $x=a/b \in K-R$ and $a \notin (b)$. How do I prove $x \notin R_M$ where $M$ is a maximal ideal containing $b$? The statement is ...
2
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1answer
40 views

A relation between homomorphisms from the polynomial ring zero on an ideal and homomorphisms from the quotient of the polynomial ring by this ideal

Let $n\geq 1$, $K$ be a field and $R\neq \{0\}$ a $K$-algebra. For Ideals $I$ and $J$ of $K[X_1\ldots,X_n]$ with $J\subseteq I$ consider $$ A(I)=Hom_{Kalg}(K[X_1,\ldots,X_n]/I,R) $$ and $$ ...
1
vote
1answer
334 views

Krull dimension in polynomial rings

Let $F$ be a field and $R=F[X_1,X_2,\ldots,X_n]$ be the polynomial ring in $n$ variables over $F$ and $P$ be a prime ideal in $R$, I'm trying to prove that$$\operatorname{ht}P+\dim R/P=\dim R$$where ...
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0answers
73 views

Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?

I am trying to find a translation of this paper either in English or French (preferably English). I am not very optimistic, but i thought of asking in case somebody is more resourceful :)
7
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1answer
177 views

Vandermonde identity in a ring

Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
12
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1answer
154 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
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0answers
26 views

A property of linearly compact module

Let $(R,\mathfrak{m})$ be a noetherian local ring, $E$ the injective hull of $R/\mathfrak{m}$, $S=\operatorname{End}_R(E)$ and $M$ a linearly compact and discrete $R-$module. Show that ...
2
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1answer
34 views

Showing that the natural map into the completion is continuous

Let $M$ be an $A$-module and $M=M_0 \supset M_1 \supset \cdots$ a sequence of submodules, which we define to be a fundamental system of neighborhoods of $0$. Thus we make $M$ into a topological group. ...
2
votes
2answers
128 views

Example of a non-free module over some Laurent polynomial ring

This is probably a naive question. What is an example of a non-free finitely generated module $M$ over some Laurent polynomial ring $$ L_n=K[X_1,X_1^{-1},\ldots,X_n,X_n^{-1}] $$ where $K$ is a field. ...
5
votes
1answer
236 views

Noetherian rings and prime ideals

Let $R$ be a noetherian ring and $P\subset Q$ be prime ideals. I'm trying to prove that if there exists another prime ideal $P_1$ such that $P\subset P_1\subset Q$ and $P\ne P_1\ne Q$, then there are ...
3
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1answer
393 views

Zero divisors and height of prime ideals in Noetherian rings.

Let $R$ be a noetherian ring, $x\in R $ be a non zero divisor, and $P$ a prime ideal of $R$ which is minimal over $(x)$. I'm trying to show that $\operatorname{ht}P=1$. Also if $Q$ is a prime ideal of ...
2
votes
2answers
201 views

Inverting formal power series wrt. composition

A formal power series $f \in R[[X]]$ is said to be invertible wrt. composition, iff there exists $g \in R[[X]]$ s.t. $f \circ g = g \circ f = X$ holds. It is easy to see, that for such $f = ...
3
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1answer
154 views

Using localization to show that any finitely generated projective module over Dedekind ring is direct sum of ideals

Hello I am stuck showing the following: If $M$ is a finitely generated projective module over a Dedekind ring $R$, then $M\cong\bigoplus_{i=1}^k\mathfrak{a}_i$ for some ideals ...
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1answer
157 views

Discrete Valuation Rings problem 2

An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying: i) $\phi(a) = \infty$ if and only if $a=0$. ii) $\phi(ab) = \phi(a) + \phi(b)$. iii) ...
6
votes
1answer
79 views

algebraic distance of an element of a ring from an ideal

Let $A$ be a commutative ring and $I$ an ideal. Does there exist a notion of "distance" of an element $x \in A$ from the ideal $I$? This "distance", need not be of the form $A\rightarrow \mathbb{R}$; ...
2
votes
2answers
207 views

Kernel of $p$-adic logarithm.

I'm completely clueless as to how to answer the following question: Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
0
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1answer
111 views

Ring of fractions problem

How do I can determine the ring of fractions of $\mathbb{Z}[X]$ ? I don't know the process that I have to follow for do it.
2
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0answers
99 views

Finitely many prime ideals lying over the same prime ideal [duplicate]

Let $A \subseteq B$ an extension of rings such that $B$ is an $A$-module finitely generated. Show that for every prime ideal $\mathfrak{p} \subseteq A$ there is only a finite number of prime ideals ...
1
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1answer
135 views

Hilbert Theorem of zeros

Use the Hilbert Nullstellensatz Theorem to prove the following result: Given $F_1, F_2, F_3 \in \mathbb{C} [X_1,\dots,X_n]$ polynomials checking the following conditions: $F_1$ is ...
2
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1answer
395 views

Commutative ring with unity Proof on the set of units?

the question is as follows (TRUE or FALSE.) If R is a commutative ring with unity, then the set of units in R forms a subring. (If true, give a short proof. If false, give a specic counter-example.) ...
6
votes
1answer
124 views

Embedding of free $R$-algebras

Let $R$ be any nontrivial commutative unital ring and $I$ and $J$ any sets with $|I|>|J|$. Does there exist an embedding of $R$-algebras $R[x_i; i\in I]\longrightarrow R[y_j;j\in J]$? When ...
2
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0answers
25 views

Analytical Independence

I am aware of the definition of analytical independence in Noetherian rings. I am wondering if anyone knows of any generalization of the concept (or similar concept ) to non-noetherian rings.
2
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1answer
163 views

Pure Submodules and Finitely Presented versus Finitely Generated Submodules

Let $A$ be a ring $M$ an $A$-module and $N$ a submodule. Definition: $N$ is called a pure submodule of $M$ if the sequence $0 \rightarrow N \otimes E \rightarrow M \otimes E$ is exact for every ...
3
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1answer
111 views

Finitely generated torsion module over a PID.

Let $A$ be a PID, $K$ be the field of fractions of $A$, and $M$ be a finitely generated torsion $A$-module. Let $M'=\text{Hom}(M,K/A)$ and $M''=\text{Hom}(M',K/A)$. I want to show that the evaluation ...
6
votes
1answer
107 views

What is the image of the map $\hom(V,V) \to \hom(\wedge^k V,\wedge^k V)$?

The title says it all. For the uninitiated: Any map $f:V \to W$ induces a map $\wedge^k V \to \wedge^k W$ by $v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so $\wedge^k(-)$ ...
10
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3answers
287 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 ...
5
votes
2answers
140 views

How to see that $\ker\left((X,Y)\otimes_R(X,Y)\to(X,Y)^2\right)\simeq k$ in $R=k[X,Y]$?

Let $k$ be a field, $R=k[X,Y]$ and $I=(X,Y)$, so that $R/I\simeq k$. I proved, using a projective resolution of $k$, that $\text{Tor}^R_2(k,k)= k$. I also proved that in general $$ ...
1
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1answer
223 views

Deduce that a Noetherian valuation ring is either a field or a Discrete Valuation Ring.

I'm trying to solve this question from a book and I have already proved 1. Let $R$ be a local domain which is not a field. Suppose that the maximal ideal $M$ of $R$ is principal and satisfies ...
2
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0answers
91 views

Integral dependence and fraction fields [duplicate]

Consider $\mathbb{Q}[x]\subset\mathbb{Q}(x)\subset\mathbb{Q}(x)[y]=:K$, where $$y^2=x,$$ and let $O_K$ be the integral closure of $\mathbb{Q}[x]$ in $\mathbb{Q}(x)[y]$. Show that ...
5
votes
2answers
284 views

Integral domains such that all proper factor rings are finite

Let $\mathbb Z$ be the ring of rational integers. If $a\in\mathbb Z$ is a non-zero element, then the factor ring $\mathbb Z/(a)$ is finite and has order $|a|$. If $\mathbb Z[i]$ is the ring of ...
7
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2answers
178 views

Nullstellensatz and the Fundamental Theorem of Algebra

I came across an interesting problem that basically said something along the lines of ``Show that Hilbert's Nullstellensatz is equivalent to the Fundamental Theorem of Algebra.'' My algebraic geometry ...
4
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1answer
97 views

quadratic extension of $\mathbb{Q}(X)$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the quadratic extension of ...
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1answer
146 views

Algebraic Curves

Let $F$ be a non-constant polynomial in $k[X_1,...,X_n]$, $k$ algebraically closed. Show that $\mathbb A^n \setminus \mathrm{V}(F)$ is infinite if $n\geq 1$, and $\mathrm{V}(F)$ in infinite if ...
2
votes
1answer
71 views

Local rings and classifying singularities

My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
-1
votes
1answer
117 views

List of examples of commutative rings [closed]

For curiosity: Can anyone present the currently known list of examples of commutative rings? As Wikipedia says, one may include polynomial rings, rings of algebraic integers and p-adic integers. What ...
5
votes
3answers
189 views

Integral closure of $\mathbb{Q}[X]$ in $\mathbb{Q}(X)[Y]$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of ...