Tagged Questions

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

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2
votes
1answer
25 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 $m-1$?
0
votes
1answer
16 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 ...
0
votes
1answer
38 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. ...
1
vote
0answers
36 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 ...
0
votes
0answers
33 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 ...
0
votes
0answers
20 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 ...
0
votes
0answers
19 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 ...
0
votes
1answer
48 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 ...
0
votes
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 ...
4
votes
0answers
81 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 ...
0
votes
0answers
37 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 $\operatorname{card}(B)=1.$ ...
0
votes
1answer
22 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} ...
3
votes
1answer
55 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 ...
2
votes
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 ...
0
votes
1answer
29 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 ...
0
votes
0answers
34 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 $f$. ...
1
vote
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
7
votes
3answers
62 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 ...
4
votes
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 ...
1
vote
2answers
36 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$.
-1
votes
0answers
26 views

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 ...
3
votes
0answers
46 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 ...
-1
votes
0answers
23 views

Reduction of every power of ideal over local ring [closed]

Let $(R,\mathfrak{m})$ be a commutative noetherian ring and $I=(a,b,c)$. Is $(a^n,b^n,c^n)$ reduction of $I^n$ for all $n$?
1
vote
0answers
35 views

What is the definition of the completion of a $\mathfrak o_K$-module at an infinite prime?

I have a number field $F$ and its ring of integers $\mathfrak o$ and an infinite place $\mathfrak p$ of $F$. Let $V$ be a finitely generated $\mathfrak o$-module. My question is, how is the ...
1
vote
1answer
27 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 ...
0
votes
0answers
38 views

Is a basic ideal normal? [closed]

Let $\def\fm{\mathfrak{m}}(R,\fm)$ be a local ring and $I$ be a basic ideal, i.e. $I$ is a reduction of itself, then $$\overline{\fm{I}^n}\cap {I^n}={\fm}I^n.$$
0
votes
0answers
45 views

Reduced associated graded rings and integrally closed ideals [closed]

Let $(R,\mathfrak{m})$ be commutative Noetherian local ring. If $\operatorname{gr}_I(R)$ is reduced, then $I^n$ is integrally closed for all $n$.
2
votes
2answers
170 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 ...
2
votes
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 ...
3
votes
0answers
65 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
1answer
44 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 ...
0
votes
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. ...
0
votes
1answer
50 views

How to check if an ideal is primary

I've the ring $A=k[x,y,z]/(z^2-xy)$ and the ideal $I=(x,y)^2$. How do I check if $I$ is prime in $A$? I know that $(x,y)$ is a prime ideal as $A/(x,y)$ is isomorphic to $k[z]$ but that does not imply ...
1
vote
2answers
54 views

Intersection of two flat submodules

Let $A$ be a ring, $M$ an $A$-module and $M_1,M_2$ two flat $A$-submodules of $M$. Is $M_1 \cap M_2$ a flat $A$-submodule of $M$?
2
votes
1answer
59 views

If $\mathfrak a\subset A$ is a finitely generated ideal, and if $\mathfrak p$ is a prime ideal, then $S(\mathfrak a)\cap\mathfrak p\not=\emptyset ?$

For an ideal $\mathfrak a\subset A,$ define $S(\mathfrak a)=\{f\in A\mid f\not\in x, \forall x\in D(\mathfrak a)\};$ namely, $S(\mathfrak a)$ is the set of elements that do not belong to any prime ...
2
votes
0answers
57 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
3
votes
1answer
93 views

On Bounded Index of Nilpotency of $R[x]$ and $M_n(R)$

A ring $R$ is said to have a bounded index (of nilpotency) if there is a positive integer $n$ such that $x^n=0$ for every nilpotent $x∈R$. Can anyone give me an example of a ring $R$ which has a ...
1
vote
1answer
38 views

Given an ideal of a ring $R$, is there any way by which the associated primes of $R/I$ can be computed without knowing a primary decomposition of $I$?

Suppose I've been given an ideal $I$ of a commutative ring $R$ and I don't know the primary decomposition of $I$. How do I find the associated primes of $R/I$? Please give some approach if possible. ...
0
votes
1answer
27 views

Inversion of an element in Picard group over commutative ring

I'm having some troubles understanding a proof in Commutative Algebra Chapter I - VII of N. Bourbaki. It's on pag 114 of the book. Here's what it says: Theorem 3 ... (ii) Conversely, if $M$ ...
0
votes
1answer
51 views

Maximal among some ideals is prime

I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy ...
7
votes
3answers
184 views

Are Dummit and Foote making a mistake in proving Cohen's theorem?

Exercise 11 on page 669 (this is Chapter 15) wants to prove Cohen's theorem that if every prime ideal of a ring is f.g. then every ideal is f.g. that is the ring is noetherian. The highbrow (perhaps?) ...
0
votes
0answers
31 views

$S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$

Behold any commutative ring $R$. Is it true that $S^{-1}R[(x_i)_{i\in I}]=(S^{-1}R)[(x_i)_{i\in I}]$ for any multiplicative subset $R$ of $S$? I couldn't find this in full Bourbaki generality, not ...
0
votes
0answers
21 views

Calculating the Hilbert polynomial of a principal ideal

If we have a field $K$, and a homogeneous polynomial $f \in R=K[x_1, \ldots, x_n]$, then the ideal generated by $f$ is a graded module over $K$, and we can calculate its Hilbert polynomial. (I am ...
-1
votes
1answer
23 views

Tensor product of the fraction field of a domain and a module over the domain

Given a fraction field $k(x)$ of the polynomial ring $k[x]$ over a field $k$ and an integral domain $R$ that is also a $k[x]$-module, is it true that $k(x) \otimes_{k[x]} R \cong Frac(R)$? I ...
0
votes
0answers
15 views

Non-closed map of spectra [duplicate]

What is the simplest example of rings homomorphism $A\rightarrow B$ such that the induced map of spectra $\text{Spec}(B)\rightarrow\text{Spec}(A)$ is not closed?
2
votes
1answer
40 views

Cohen-Macaulay ring and module: R-regular vs M-regular

Let $R$ be a Cohen-Macaulay ring and $M$ be a finite generated maximal Cohen-Macaulay module. I know that the R-regular sequence must be $M$-regular. Here are my questions: 1) Must an $M$-regular ...