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

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2
votes
1answer
98 views

Correlation between localisation & local rings

If $R$ is a commutative ring and $S$ a multiplicative subset of $R$, then one can define the localisation $S^{-1}R$ of $R$ at $S$. Now if $p$ is a prime ideal of $R$ and we set $S=R\setminus p$ then ...
3
votes
1answer
392 views

Nice proof for finite of degree one implies isomorphism?

Let $f: X \longrightarrow Y$ be a morphism of varieties over $\mathbb{C}$ and assume it is finite of degree 1, i.e. it is surjective and $$ [K(Y):K(X)] = 1 \quad \quad (*) $$ i.e. the function fields ...
2
votes
3answers
152 views

Nullstellensatz in the coordinate ring $\Gamma (X)$

One of the many statements of the Hilbert's Nullstellensatz is the following: If $k$ is an algebraic closed field, and $\mathfrak a$ is an ideal of the ring $k[T_1,\ldots,T_n]$ then $I(V(\mathfrak ...
5
votes
1answer
147 views

Artinian affine $K$-algebra [duplicate]

Let $K$ be a field and $A$ an affine $K$-algebra. Show that $A$ has (Krull) dimension zero (is artinian) if and only if it is finite dimensional over $K$.
1
vote
1answer
119 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
143 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
votes
0answers
64 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 ...
1
vote
0answers
55 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.
6
votes
2answers
245 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
votes
0answers
38 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
120 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
228 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
votes
1answer
59 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
votes
1answer
168 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
52 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
votes
1answer
105 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 ...
8
votes
1answer
434 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
votes
2answers
66 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
votes
1answer
44 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 $$ ...
2
votes
1answer
695 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 ...
5
votes
0answers
79 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 :)
8
votes
1answer
380 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
votes
1answer
167 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: ...
1
vote
0answers
28 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
votes
1answer
48 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
223 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. ...
7
votes
1answer
453 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
votes
1answer
689 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
469 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
votes
1answer
184 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 ...
0
votes
1answer
233 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
95 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
413 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
votes
1answer
124 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
votes
0answers
104 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
votes
1answer
150 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
votes
1answer
656 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
133 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
votes
0answers
31 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
votes
1answer
300 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
votes
1answer
149 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
117 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 $\wedge^k(-)$ ...
11
votes
3answers
434 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
161 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
vote
1answer
381 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
votes
0answers
93 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
407 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
votes
2answers
304 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
votes
1answer
106 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 ...
-1
votes
1answer
181 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 ...