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

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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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1answer
45 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 ...
3
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1answer
51 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 ...
2
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1answer
67 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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1answer
57 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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30 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 ...
2
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1answer
78 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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2answers
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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0answers
39 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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1answer
15 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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2answers
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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1answer
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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0answers
31 views

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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1answer
36 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 ...
2
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1answer
35 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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1answer
37 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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2answers
167 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 ...
1
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1answer
39 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 ...
2
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2answers
71 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 ...
2
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1answer
48 views

How to show that $\mathbb Z+x \mathbb Q[x]$ is a GCD domain?

How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain, that is, the sum of two principal ideals is again a principal ideal ? Or at least, how to show that it is a GCD domain ? (This will then ...
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1answer
44 views

Looking for an example of a GCD domain which is not a UFD

I know that every UFD (unique factorization domain) is a GCD domain i.e. g.c.d. of any two elements, not both zero, exists in the domain. I am looking for an example of a GCD domain which is not ...
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2answers
45 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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25 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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1answer
66 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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0answers
54 views

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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1answer
30 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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1answer
40 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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0answers
36 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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1answer
27 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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0answers
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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 ...
2
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1answer
90 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
37 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} ...
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1answer
44 views

Is $\mathbb{Q}_p$ a $\mathbb{Z}_p$-algebra of finite type?

Let $p$ be a prime. The p-adic numbers $\mathbb{Q}_p$ are an algebra under the $p$-adic integers $\mathbb{Z}_p$ via the localization $\mathbb{Z}_p\to \mathbb{Z}_p[\frac{1}{p}]=\mathbb{Q}_p$. Is ...
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1answer
30 views

Right-exactness of Kähler-Differential and zeroth relative homology functor

In Commutative Algebra: with a View Toward Algebraic Geometry Eisenbud describes the Kähler-Differential as a functor that assigns $\Omega_{S/R}$ to an $R$-Algebra $S$ and to a commutative diagramm $$ ...
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28 views

Maximal $R$-sequences in non-noetherian rings

If $(a_1,...a_n)$ is a maximal $R$-sequence in an ideal $I$ of a noetherian commutative ring $R$ it is seen that $I⊆∪P$, where the union is taken over all the associated prime ideals of the ...
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1answer
75 views

Flatness under reduction

Suppose that $f : X \to Y$ is a flat morphism of schemes. Is $f_\text{red} : X_\text{red} \to Y_\text{red}$ necessarily flat? Are there any hypotheses that would guarantee this?
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1answer
34 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let ...
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0answers
71 views

Commutative algebra: Integral extension [closed]

Let $R ⊆ S$ be an extension of rings, and let $u$ be a unit in $S$. $(i)$ Prove for $α ∈ R[u] \cap R[u^{−1}]$, there is $n > 0$ such that $αA ⊆ A$, where $$A =\langle 1, u, u^2, . . . , u^n ...
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0answers
21 views

Normalization of curve

How do to normalize the curve $ax^2+y^2=1+bx^2y^2$ (hard exercize)? I tried the substitution $t=xy$, $u=y$. But I get $at^2/u^2+u^2=1+bt^2$ can i multiply it on $u^2$.
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1answer
59 views

How to show that an object is a discrete valuation ring? (Fulton, Exercise 2.14)

I need some help to solve the following problem that appears on page 31 of the book of William Fulton entitled Algebraic Curves. Exercise : Let $ V = \mathbb{A}^1 $, $ \Gamma (V) = k[X] $, $ K = ...
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1answer
37 views

Associated primes of an $R$-module

An associated prime of an $R$-module $M$ is an ideal of the form $Ann_R(N)$ where $N$ is a prime sub-module of $M$ in the sense that $N$ is nonzero and $Ann_R(N)=Ann_R(N')$ for each nonzero sub-module ...
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1answer
31 views

Maximal ideals of commutative Artinian rings

I would like some help on an exercise I thought I had done correctly at first glance, but obviously have doubts about. The question is; Let $R$ be a commutative Artinian ring. Then R has finitely ...
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30 views

Working with affine Varieties

Hi guys I just wanted to hear some input on this, $A=(x^4+y^4-1)$ and $B=(p^2+w^2-1)$ are affine varieties in $C^2$. We want to show that if we apply the map $f(a,b)=(a^2,b^2)$ then $f(A) \subset ...
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1answer
68 views

About the $k$-subalgebras of $k[x]$

Still in my "commutative algebra marathon", I came across the following exercise: Any $k$-subalgebra $A$ of $k[x]$ is finitely generated as $k$-algebra; also, if $A\ne k$, then $\dim A=1$. ...
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0answers
35 views

Is smoothness of $X\to Y$ for noetherian $X$ a local property on $X$?

Let $X$ be a noetherian scheme. If $X$ is regular, then the scheme $\operatorname{Spec}(\mathcal{O}_{X,x})$ is regular for all points $x\in X$. I wonder if something analog is true for smoothness of a ...
4
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0answers
76 views

Property of free submodules for a module over a PID

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 not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
0
votes
1answer
38 views

Decomposition of a maximal ideal as a union of smaller prime ideals

Let $K$ be a field, $S=K[X,Y]$ the polynomial ring in two variables and consider the ideal $M=\langle X,Y\rangle$ (ideal generated by $X$ and $Y$). Show that $M$ is a union of strictly smaller prime ...
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vote
3answers
91 views

How to show that the ideal $(X^{3},XY,Y^{n})$ of $K[X,Y]$ is primary?

I'm working on a problem in Sharp's Steps in commutative algebra, to be precise exercise 4.28 which states the following: Let $K$ be a field and $R = K[X,Y]$ be the polynomial ring in the ...
4
votes
2answers
77 views

Annihilator of a maximal ideal in a ring

Let R be a ring and M a maximal ideal in R. Prove or disprove: If M is contained in the set of zero divisors of R, then ann(M) is not 0. It is easy to see that the statement is true when M is ...