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

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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 ...
1
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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 ...
0
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1answer
17 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
57 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 ...
3
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1answer
40 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
33 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 ...
0
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1answer
37 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
40 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 ...
0
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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
174 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
77 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
54 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 ...
1
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1answer
47 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 ...
1
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2answers
48 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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0answers
26 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 ...
1
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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 ...
1
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0answers
57 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) ...
1
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1answer
35 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
42 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
37 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 ...
1
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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
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 ...
2
votes
1answer
94 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 ...
1
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1answer
52 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 ...
2
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1answer
39 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} ...
1
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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 ...
5
votes
1answer
32 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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0answers
29 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 ...
5
votes
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?
2
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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
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
63 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
38 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
34 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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0answers
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 ...
2
votes
1answer
70 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 ...
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0answers
77 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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3answers
98 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
83 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 ...
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0answers
43 views

Covering of $\mathbb{P}^n$ and the complement of a point

Let $p$ be a closed point in $\mathbb{P}^n$ for some integer $n$ and $\{U_i\}$ be an affine open covering of $\mathbb{P}^n\backslash p$. Does there exists an open set in the covering, say $U_0$ for ...
1
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1answer
63 views

Every prime is maximal in a Jacobson ring?

In Attiyah commutative algebra page 71, it is given some equivalent definitions of Jacobson ring. One of the definitions are that every prime ideal which is not maximal is equal to the intersection of ...
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2answers
83 views

There's no surjective ring homomorphism from $\mathbb{Z}[x_1,\dots,x_n]$ onto $\mathbb{Q}$.

I'm trying to prove that, if a field $A$ is also a finitely generated $\mathbb{Z}$-algebra, then $A$ is finite. The proof I found for this depends on the fact that $\mathbb{Q}$ cannot be a finitely ...
0
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1answer
39 views

Finding integral dependence for polynomials [closed]

How can I find a integral dependence for each $f\in K[x]$ over $K[x^2]$? For arbitrary field $K$.
4
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1answer
36 views

A noetherian local ring having a height one principal prime is a domain

$A$ is a commutative ring with with $1$. If $A$ is a Noetherian and local ring and $A$ has a principal prime ideal of height $1$ then show that $A$ is a domain. Can anybody give some hint.I tried ...
0
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0answers
43 views

Locally free sheaf on Cohen-Macaulay scheme and Serre's criterion

Let $X$ be a projective locally Cohen-Macaulay scheme and $\mathcal{F}$ be a locally free sheaf on $X$. If I understand correctly the definition of Serre's criterion $S_k$, $\mathcal{F}$ satisifies ...
3
votes
1answer
72 views

Is $k[x][[h]]$ finitely generated as $k[[h]]$-algebra?

Is $k[x][[h]]$ finitely generated as an algebra over $k[[h]]$, where $k$ is a field, and $xh=hx$.