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

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How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
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17 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
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25 views

Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the ...
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1answer
19 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
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41 views

One-dimensional Noetherian UFD is a PID

Hi I am looking for a reference which has a self-contained (elementary; that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does ...
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45 views

Maximal ideal in a polynomial ring over a field that is not algebraically closed

I want to prove that although $K$ is a field that IS NOT algebraically closed, every maximal ideal in $K[x_1, \ldots, x_n]$ can be generated by $n$ elements. To prove this, I am following the next ...
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26 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
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33 views

Contraction of a maximal ideal in a polynomial ring

I have two questions: If $K$ is a field, $R=K[x_1,\ldots,x_n]$, the ring of polynomials over $K$ with $n$ indeterminates, and $M$ is a maximal ideal of $R$ why is the contraction $N$ of $M$ to ...
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62 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
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55 views

Question on a property of $\mathrm{Ass}(M)$ for modules over noetherian rings

I got stuck reading a proof of the following lemma (Lemma 0.19 in this file): Lemma Suppose that $M$ is a module over a commutative noetherian ring $R$ and let $m\neq 0 \in M$. Let $S$ be a ...
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58 views

If the localizations of two submodules with respect to any prime ideal are equal then the submodules are equal [on hold]

I want to prove the following: Let R be a commutative ring with 1 and let N and L be two submodules of an R-module M. If the localizations of N and L with respect to any prime ideal of R are ...
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57 views

Localization of a regular local ring is regular

Quoting Hartshorne's Algebraic Geometry Definition. We say a scheme $X$ is regular in codimension one if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular. The most ...
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40 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
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20 views

Module length and connection between $\varphi,\det\varphi$ [on hold]

$e_A(\varphi,M) = l_A(\mathrm{coker}\varphi) - l_A(\ker\varphi)$ Let $A$ be domain. I want to prove that $e_{A}(\varphi,A^n) < \infty \iff e_{A}(\det\varphi,A) <\infty$ using the fact that ...
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35 views

Relatively prime ideals in Dedekind Domains

I am currently working through Lang's Algebra and have come across an exercise I can not solve (Chapter II, Exercise $19$). Any help would be greatly appreciated. Let $R$ be a Dedekind domain. ...
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Talking about varieties

hi I was recently reading ideals varieties and algorithms. I ah having problems showing things are not affine varieties. Previously with problems like. $V= \{ (a,a) | a \in R^* \}$ it was much easier ...
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70 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
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45 views

Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
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41 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
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40 views

Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words, if the associated prime ideals of $R/I$ are the minimal prime ideals ...
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45 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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65 views

How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
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54 views

If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
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29 views

Subvarieties and finding ideals

Hi guys I am stuck working on this problem. I have a surface $W= V(xz-y^2)$ and we are trying to find an ideal $J \in K[W]$ so that the $V_w(J)=V(y-x^2,z-x^3)$ I showed that the second thing which is ...
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1answer
77 views

Functorial construction with two integral domains

Motivated by this question: Let $\mathsf{Int}$ be the category of integral domains with ring homomorphisms (perhaps only injective ring homomorphisms, if you need this). Is there a functor ...
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38 views

Residue class field of coordinate ring

If $X$ is an irreducible affine curve over an algebraically closed field $k$, then its coordinate ring $O(X)$ is a Dedekind domain. Suppose $\mathfrak{p}$ is a prime (hence maximal) ideal in $O(X)$ ...
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35 views

Property of free submodules for a module over a PID [duplicate]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
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14 views

Residue class ring of Dedekind domain

Zariski and Samuel Commutative Algebra Ch V para 7 makes the following statement: If $R$ is a Dedekind domain with an ideal $\mathfrak{a}=\prod_i\mathfrak{p}_i^{n(i)}$ factored into prime ideals, ...
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52 views

Module structure of base extension via tensor product

Let $A,B$ be commutative rings. Defining a product of $B\otimes_{A}B$ as $(b_1 \otimes b_2)\cdot (b_3 \otimes b_4)=(b_1b_3)\otimes(b_2b_4)$, this becomes a commutative ring. Defining $b\cdot(b_1 ...
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37 views

Question on the existence of a prime ideal contained in the $\ker$ of a homomorphism $\mathbb{C}[x,y]\rightarrow\mathbb{C}[t]$.

I found this exercise in a basic algebraic geometry book: Let $f:\mathbb{C}[x,y]\rightarrow \mathbb{C}[t]$ a non-zero homomorphism such that $\ker f$ strictly contains a prime ideal $P\neq0$. Is it ...
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tensor product of formal power series

Let $A[[h]]$ be the formal power series algebra over $\mathbb{C}[[h]]$, here $\mathbb{C}$ is the complex number field. Is the canonical map $A[[h]] \otimes_{\mathbb{C}[[h]]} A[[h]] \to ...
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35 views

A comparison between heights and between grades

I search for noetherian commutative rings having distinct prime ideals $P⊂Q$ with no primes between them, where $grade(Q)≠grade(P) +1$, or $height(Q)≠height(P)+1$. If $R$ is Cohen-Macaulay, are the ...
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30 views

Gröbner Basis and Division Algorithm

I recently read a lemma on a course in Commutative Algebra that states, If $G$ is a Gröbner Basis for an Ideal $I$ in $k[x_{1},x_{2}...x_{n}]$, then a polynomial $f$ belongs to $I$ if and only if ...
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35 views

Describing ideal that vanishes at the variety

We have the following morphism $$\phi(a_1,..a_m;b_1,...,b_n)= \begin{pmatrix} a_1 b_1 & \ldots & a_1 b_n \\ \vdots & \ddots & \vdots \\ a_mb_1 & \ldots & a_m b_n ...
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161 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
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38 views

Element in no prime ideal $\iff$ it is a unit

I was working through Atiyah & MacDonald, chapter 1 section 1 problem 17 part iii) where it says Let $R$ be a ring and $f\in R$. Define $V(f)$ to be all elements of ...
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65 views

Reference to complete proof that integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$?

Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the ...
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43 views

Prime Ideals and multiplicative sets

I am currently studying a course on commutative algebra and came across this statement: An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set. I have proved ...
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23 views

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Can one compute $\ell_S(S/(a_1^i,a_2^j))$?

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Let $x_i$ be the natural image of $X_i$ in $S$. Set $a_1=x_1+x_3$ and $a_2=x_2+x_3$. $a_1,a_2$ is a system of parameters of $S$. So ...
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63 views

In any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated; can it be prove without A.C.?

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring ...
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Polynomial division, multivariable, indeterminates

Trying to understand something in the proof of Nullstellensatz, if we have a polynomial $p(x_1,...,x_n,t) \in k[x_1,...,x_n,t]$ with $f(t)$ divides $p(a_1,...,a_n,t)$ for all fixed $(a_1,...,a_n) ...
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29 views

$X_1,X_2$ disjoint closed in $Spec(R)$ properties

This is a problem in three parts, I managed to prove the first part, but the others I couldn't. Let $R$ be a ring and let $X_1,X_2\subset Spec(R)$ be closed (in Zariski topology) and disjoint such ...
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39 views

If $I$ proper ideal of $R$, $S$ ring extension of $R$, and $u$ a unit in $S$, then $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$

Let $R ⊆ S$ be an extension of rings, and let $u$ be a unit in $S$. Let $I$ be an ideal of $R$ with $I \ne R$. Show that $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$. Here is what I try: I have ...
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32 views

Finite union of algebraic affine varieties

I'm studying Commutative algebra and Algebraic Geometry. I have proved the following proposition: $V(I)∪V(J)=V(IJ)=V(I∩J)$ I would know why sometimes is better to use the product of ideals instead ...
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33 views

noether normalization and complete intersection

Let I be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
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26 views

A normality criterion

I'm trying to solve exercise 8.5 from the book "A course in Commutative algebra" by Gregor Kemper. It says Let $R$ be a ring and $a\in R$ such that $a$ is not a zero divisor, the ideal $(a)$ is a ...
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34 views

Is Spec R compact? [duplicate]

And if so, why? I'm having some trouble with this. I know that the $D_{f}$ (set of primes not containing $f$) are the open sets and form a basis for the Zariski topology; but I do not know how to go ...
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1answer
74 views

An integral domain with Krull dimension 1 which is neither Noetherian nor integrally closed

It seems like a common exercise to try and find rings which only satisfy some of the conditions in the definition of a Dedekind domain. Rings that satisfy exactly 2 of the three conditions were very ...
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1answer
51 views

If $a$ is algebraic over $\mathbb Z$, every polynomial in $a$ can be expressed in a low degree?

I ran into something when working on a problem in Artin's Algebra. If $a$ is algebraic over $\mathbb Z$ with order $n$ (i.e., the smallest degree integer polynomial with $a$ as a root has degree ...
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1answer
36 views

“Direct sums of injective modules over Noetherian ring is injective” and its analogue

I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} ...