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

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19 views

$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
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1answer
34 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
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1answer
22 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
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17 views

Why does the following reduction holds? [on hold]

Let $I=(a,b,c)$ be an ideal of local ring $R$. Then $(a^n,b^n,c^n)$ is a reduction of $I^n$ for all $n$
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19 views

Why does the following length holds? [on hold]

Let $R=k[x,y]$, $0=a_n<a_{n-1}<...<a_1$ and $0=b_1<b_2<...<b_n$ be integers. Set $I=(x^{a_i}y^{b_i}: i=1,2,...,n)$. Why $\ell(R/I)=\sum_{i=1}^{n-1}a_i(b_{i+1}-b_i)$
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18 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
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1answer
30 views

Question about notation on ideals

If $R$ is a commutative ring and $a,b \in R$ then $(a)+(b)=(a,b)= \{xa+yb : x,y \in R \}$, however if $I$ is an ideal of $R$ then what is $(I,a)$? My guess is $(I,a)=\{hg + xa : h,x \in R, g \in I ...
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1answer
20 views

Comparing an ideal and its saturation

Let $S = k[x_0,x_1,\ldots,x_n]$ with its usual grading and let $I \subset S$ be a homogeneous ideal not containing $S_+ = (x_0,x_1,\ldots,x_n)$. We define the saturation of $I$ to be the homogeneous ...
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23 views

proof of Proposition 3.3.18 in Bruns and Herzog

This set of questions pertains to the proof of Proposition 3.3.18(b) in Bruns and Herzog, Cohen-Macaulay Rings: Question 1: It seems to me that under the hypothesis (a) of the theorem, the ...
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42 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
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1answer
46 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
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1answer
32 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...
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1answer
30 views

Example of Gorenstein local ring of dimension 1

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
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1answer
36 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
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36 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
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1answer
17 views

Krull dimension and zero divisors of $k[x,y,z]/(x^ay,x^bz)$

I found the primary decomposition of $(0)$ in the ring $k[x,y,z]/(x^ay,x^bz)$, where $a\geq b \geq 1$, $k$ is alg. closed, to be $(x^b) \cap (x^a,z) \cap (y,z)$ (is this correct?). Now I am now ...
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1answer
26 views

Presentation of a local complete intersection

What is the simplest example of a local (noetherian) complete intersection ring $R$ that can not be presented as $R=S/I$, where $S$ is a regular local ring and $I$ is an ideal generated by a regular ...
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1answer
32 views

closed and open subscheme of affine scheme

Let $X=Spec(A)$ be a noetherian affine scheme. Let $I_1, \ldots, I_n$ be ideals of $A$ such that $I_i + I_j = 1$ for all $i \neq j$. Define $X_i = Spec(A/I_i)$ so that X is the disjoint union of the ...
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1answer
33 views

Dimension of irreducible variety

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?
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30 views

Relation about prime ideals in $B$ and invariant subring $B^G$

Suppose $B$ is a commutative ring, $G$ is a finite group acting on $B$, $A=B^G$ is the invariant subring. Suppose $P$ is a prime ideal in $A$, $Q_1,...,Q_s$ are all the prime ideals in $B$ such that ...
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1answer
43 views

Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
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0answers
45 views

When is a holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
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1answer
44 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
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24 views

On different versions of Schwarz Zippel

Theorem (Schwartz, Zippel). Let $P\in F[x_1,x_2,\ldots,x_n]$ be a non-zero polynomial of total degree $d≥0$ over a Field $F$. Let $S$ be a finite subset of $F$ and let $r_1,r_2,...,r_n$ be selected at ...
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0answers
21 views

Find some differential operators in the D-modules theory

I begin with the algebraic $D$-modules, and here are my questions: 1) What is the ring of $\mathbb C$-linear differential operators on $\mathbb C[[x]]$? 2) Let $S=k[s,t,s^{-1},t^{-1}]$. Prove ...
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1answer
50 views

How is this method of finding a maximal ideal specific to finite algebras over a field?

Let $A$ be a finitely generated $K$-algebra over a field $K$. A typical problem is to find a maximal ideal $\frak{m}$ such that $f\notin\mathfrak{m}$ and it does not coincide (or contains) another ...
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1answer
56 views

Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...
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83 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
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38 views

Cardinal of a linearly independent subset of $R$-module

Let $R$ be a commutative ring, and consider $R$ as an $R$-module with the action given by the product of $R$. Prove that if $B\subset R$ is linearly independent, then $\operatorname{card}(B)=1.$ ...
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1answer
23 views

Open and closed sets for j-Spec $A$.

The following is from Matsumura, Theorem 4.10 Let $A$ be a ring and $M$ a finite $A$-module. (i) For any non-negative integer $r$ set $$U_r = \{p \in \text{Spec} \space A | M_\mathfrak{p} ...
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1answer
57 views

Counterexample - modules over non-Noetherian domain

Does anyone know an example of a (necessarily non-Noetherian) domain $A$ and a finitely generated $A$-module $M$ with the property that $M_f$ is not free for any nonzero $f \in A$? This would provide ...
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1answer
33 views

Monomial ideal is radical iff it is generated by square-free monomials

I'm trying to prove that if $ K$ is a field and $ I $ is a monomial ideal in $ K[x_1, \dots, x_n] $, then $$\sqrt{I} = I \iff I ~\text{is generated by square-free monomials}$$ So I tried to do the ...
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1answer
30 views

Monomial ideal as a vector space

I'm to prove the following statement: Let $ K $ be a field. And ideal $ I $ in $ K[x_1, \dots, x_n] $ is monomial (generated by monomials in $ x_1, \dots, x_n $) $ \iff $ it is spanned on monomials ...
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36 views

Integral points of proper rational functions

Let $f \in \mathbb{Q}(X_1,\dots,X_n)$ be an arbitrary rational function which is not a polynomial, and let $D = \{ x \in \mathbb{Z}^n : f(x) \in \mathbb{Z} \}$ be the set of integral points of $f$. ...
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1answer
50 views

Modules of Finite Length over Local Artinian Rings

Let $R$ be a commutative local artinian ring with identity. Denote its maximal ideal by $\mathfrak{m}$ and let $\mathbb{k}$ denote the residue field $\mathbb{k}=R/\mathfrak{m}$. Assume also that there ...
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55 views

question on ideals in rings

Let $S=K[x_1,\dots,x_n]/J$ be a ring where $K$ is a field of characteristic $0$ and $J$ is an ideal with $Z(J)$ being the zero set of the ideal. For every $\tilde{q}\in S$, let $q$ be the lowest ...
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1answer
37 views

Interpretation of hint for Exercise 2.19b of Eisenbud

I am doing exercise 2.19b of Eisenbud's Commutative Algebra with a View Towards Algebraic Geometry. Here we have an $R$-module $M$ and elements $\{f_i\}$ which generate the unit ideal. The exercise ...
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3answers
63 views

What are the irreducible components of $V(xy-z^3,xz-y^3)$ in $\mathbb{A}^3_K$?

What are the irreducible components of the algebraic set $V(xy-z^3,xz-y^3)$ in $\mathbb{A}^3_K$? Here I"m just letting $K$ be an algebraically closed set. Normally, what I do is take the equations ...
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1answer
57 views

Characterization of free modules

Let $M$ be a finitely generated module over a commutative ring $A$. Is it true that if there exists a positive integer $n$ and a pair of homomorphisms $\pi:A^n\rightarrow M$ and $\phi:A^n\rightarrow ...
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2answers
37 views

ideal in the ring of smooth functions

What is an ideal $I$ of the ring of smooth functions $C^{\infty}(\mathbb R)$ which is not finitely generated and for all $x\in\mathbb R$ there exist $f\in I$ such as $ f(x)\neq 0$.
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Graded rings and Noetherian rings

It is true that given a graded ring $R$, it is Noetherian if and only if $R_0$ is Noetherian, and $R$ is finitely generated as an $R_0$-algebra. Is there a nice counterexample where $R_0$ is ...
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47 views

Power series modulo polynomials

I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon. $\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and ...
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1answer
30 views

Semi-simple commutative algebra

Let $A$ be a semi-simple commutative algebra over a field $F$, and $F$ is algebraically closed. The proposition is that we can express $A=Fe_1 \oplus ... \oplus Fe_n$, where $e_i$ are orthogonal ...
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2answers
174 views

Ideals-algebraic set

Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3) = \{ (t,t^2,t^3) \mid t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3)) = \langle Y-X^2,Z-X^3 \rangle$$ Finally, prove that the ...
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1answer
48 views

Density of maximal spectrum

It's well known that for algebraically closed field $k$ maximal spectrum of finitely generated $k$-algebra is everywhere dense in whole spectrum of this algebra. What can be said in the case of ...
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68 views

A certain natural map between Tor functors

Consider the following Here $A$ is a flat (commutative, unital) $k$-algebra ($k$ a commutative ring) and $\mu:A\otimes_k A\rightarrow A$ is by $\mu(a\otimes b)=ab$, $\mathcal{M}$ denotes a maximal ...
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1answer
52 views

Basic algebraic geometry question (confused about conventions)

I am completely new to algebraic geometry so please bear with me. I have started going through James Milne's notes as a first reference and have been finding them quite good. I have now turned to ...
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1answer
36 views

Simultaneous congruences

Let $\mathbb K$ be a finite field and $\mathbb K[x, y]$ the polynomial ring in the commuting indeterminates $x$ and $y$. Consider the factor ring $\mathbb K[x, y]/\langle x^3, y^3\rangle $. Can we ...
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1answer
45 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
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1answer
19 views

Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible

I have the following situation: Let $B \subseteq B'$ be a ring extension such that $\text{Quot}(B) = \text{Quot}(B') =: K$ and $\text{dim}(B) = \text{dim}(B') = 1$ where $B'$ is a Dedekind domain. ...