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

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1answer
344 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 ...
4
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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
184 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
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2answers
137 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
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1answer
247 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
440 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
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2answers
207 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
161 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
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1answer
80 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
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2answers
214 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 ...
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1answer
113 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.
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0answers
100 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 ...
0
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1answer
137 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
400 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
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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 ...
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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
171 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
112 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
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1answer
108 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 $\wedge^k(-)$ ...
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3answers
294 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
229 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
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2answers
288 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
184 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
149 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
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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
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1answer
121 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
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3answers
190 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 ...
0
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1answer
51 views

Kernel of canonical morphism in inductive limit (proof by induction)

Let $\langle I, \leqslant \rangle$ be a directed poset and $\langle M_i, \mu_{i,j} \rangle$ be a directed system of $A$-modules over $I$. Now let $$ C = \bigoplus\limits_{i \in I} M_{i}, $$ and $D$ ...
3
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1answer
578 views

Jacobson radical equal to nilradical in $R[X]$

Let $R$ be a non-zero commutative ring with identity. Let $\textrm{nilrad}(R)$ be the nilradical of $R$, which can be characterised either as the intersection of all prime ideals of $R$, or as the ...
6
votes
1answer
114 views

Injective hull and some Hom

Let $R$ be a commutative ring with unit. Suppose $P\in Spec(R)$ and let $E=E(R/P)$ be the injective hull of $R/P$. What can we say about $Hom_R(R/P, E)$. We know that $R/m\cong Hom_R(R/m, E)$, where ...
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1answer
85 views

Commutative Algebra: nilpotent elements.

Let $f=\sum_{n=0}^{\infty}a_nx^n$. If $f$ and $a_0$ is nilpotent how I can prove that $f-a_0$ is nilpotent? Or if $f^n=0$ and $a_0^n=0$ how can I prove that $(f-a_0)^n=0$, where $n\in \mathbb N$.
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1answer
201 views

Show that a map is continuous in the Zariski topology

Let $R,S$ be two commutative rings with unity and $\alpha :R\to S$ be a ring homomorphism, for $f\in R$ is a non nilpotent element let $R_f$ denote the localization of $R$ with respect to the ...
6
votes
2answers
227 views

Submodules of a free module over a commutative ring

Let $R$ be a commutative unital ring, $I$ a set, and $R^{(I)}$ the free module on $I$. Can there be a submodule $R^{(J)}\cong M\leq R^{(I)}$ with $|J|\!>\!|I|$? Can $R^{(I)}$ be generated (as a ...
2
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1answer
145 views

Spec R is irreducible

A topological space is called reducible if $X=X_1\cup X_2$ for two closed subsets $X_1,X_2$ with $X_1\ne X\ne X_2$. Otherwise its called irreducible, want to show that $\text{Spec}(R)$ is irreducible ...
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0answers
39 views

Conntectedness of SpecR [duplicate]

Let $R$ be a commutative ring with unity, $e\in R$ is called and idempotent if $e^2=e$ and if $e\notin \{0,1\}$ then it is called a non-trivial idempotent.want to show that $\text{Spec}R$ is not ...
0
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1answer
87 views

Monomial ordering problem

I've got the following problem: Let $\gamma$, $\delta$ $\in$ $\mathbb R_{> 0}$. The binary relation $\preceq$ on monomials in $X,Y$ is defined: $X^{m}Y^{n} \preceq X^{p}Y^{q}$ if and only if ...
5
votes
1answer
80 views

hypersurface intersected with generic line

Let $f \in \mathbb{R}[x_1,\cdots,x_m]$ be a homogeneous multivariate polynomial of degree $n$. Now, for $u,v \in \mathbb{R}^n$ the form $f(\lambda u + \mu v)$ can be written as $\mu^n h(\lambda/\mu)$, ...
1
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1answer
82 views

Flatness of Formal Power Series

According to Matsumura's Commutative Ring Theory, Ex. 7.4, the direct product of flat modules over a Noetherian ring $A$ is flat. How can we use this result to conclude that the formal power series in ...
2
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1answer
263 views

Direct Sum/Product of Flat Modules

Let $\left\{M_{\lambda}\right\}$ be a family of flat $A$-modules and define $M = \bigoplus_{\lambda} M_{\lambda}$. Let $0 \rightarrow N' \rightarrow N$ be an exact sequence of $A$-modules. Tensoring ...
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1answer
66 views

Localization of a ring which is not a domain

Let $A$ be a ring (commutative with $1$), let $S$ be a multiplicatively closed subset of $A$, i.e $S$ is contained in $A$ , $1\in S$ and $a,b\in S$ implies $ab\in S$, for every $a,b\in A$. Consider ...
2
votes
1answer
180 views

localization of a module and annihilators

I've just started reading on my own about localizations of modules. I've run into a difficulty as follows: Let $R$ be a commutative ring with unity, $M$ an $R$-module, $S\subseteq R$ a multiplicative ...
16
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3answers
679 views

If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ ...
3
votes
1answer
108 views

Computing kernel of ring homomorphism

I am trying to answer the question already asked here. My question is two parts: 1) I think I have found a proof on my own, could someone check it is valid? Modulo that ideal, $x_i\equiv a_i$ so ...