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

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3
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1answer
37 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 ...
1
vote
1answer
38 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
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 $f$. ...
1
vote
1answer
57 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
64 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 ...
3
votes
1answer
65 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
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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votes
0answers
29 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
48 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
vote
1answer
31 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 ...
1
vote
2answers
182 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
72 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
53 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
39 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 ...
2
votes
1answer
53 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 ...
1
vote
1answer
21 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
56 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 ...
0
votes
2answers
60 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
73 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 ...
2
votes
1answer
101 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 ...
0
votes
1answer
40 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
54 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 ...
8
votes
3answers
216 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
27 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
34 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
18 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?
3
votes
1answer
44 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 ...
0
votes
1answer
41 views

On Prime and Maximal Ideals in a Commutative Ring with Unity

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal ...
5
votes
2answers
86 views

Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)?

Let $R$ be a local ring containing a field isomorphic to its residue field $k$. Assume $R$ is a localization of a finitely-generated $k$-algebra. Can $\Omega_{R/k}$ be free of rank $r\neq\dim{R}$? ...
0
votes
1answer
45 views

Dedekind domain necessary for equivalence of flatness and torsion-free

It is well-known that for finitely generated modules over a Dedekind domain, flatness and torsion-free are equivalent. Is this true for general Noetherian rings? If not, where is the dimension one ...
3
votes
2answers
77 views

Product of ideals for Nakayama's Lemma

The result to be proved is the following: Let $R$ be a local Noetherian ring. Then the minimum number of generators of the unique maximal ideal $P$ equals the dimension of $P/P^2$ as a vector space ...
0
votes
1answer
36 views

System of polynomial equations and Nullstellensatz

Let $k$ be an algebraically closed field and the field $K$ contains $k$. I am trying to prove that if $F_1,...,F_m\in k[x_1,...,x_n]$ and the system of polynomial equations $F_1=0,...,F_m=0$ has the ...
1
vote
1answer
116 views

Integral closure in field of fractions.

Let $I$ be the ideal generated by $2xy+x^2+y^3$ in $\mathbb{R}[x,y]$. Define $A:=\mathbb{R}[x,y]/I$, I want to find the normalisation of $A$, that is, the set $B= \{ a \in \text{Frac} A : \text{a ...
1
vote
2answers
114 views

Commutative ring is semisimple iff it's isomorphic to a finite direct product of fields.

I am trying to prove the following: Let $R$ be a commutative ring. Prove that $R$ is semisimple if and only if it is isomorphic to a direct product of a finite number of fields. Suppose $R$ is a ...
4
votes
0answers
48 views

a subtle detail in the proof of Theorem 3.3.7 of Bruns and Herzog

Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ ...
2
votes
1answer
26 views

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $rad A =rad B$

If $A$ is a semilocal ring and $f:A\rightarrow B$ is a surjective homomorphism, then $f(rad A) =rad B$ I know that if A is a semilocal ring and if $I_{1},\cdots, I_{n}$ are all of its maximal ideals, ...
-1
votes
1answer
73 views

Maximal nor prime ideal [closed]

Let $R = C([0,1])$ be the set of continuous functions from $[0,1]$ to $\mathbb R$. Consider $R$ as a ring with the following operations $(f + g)(x) := f(x) + g(x) $ and $(f.g)(x) := f(x)g(x)$ Show ...
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vote
0answers
40 views

Is there any relationship between localization and completion of a module?

Let $R$ be a commutative ring, $\mathfrak p$ a prime ideal of $R$ and $M$ an $R$-module. I've seen the terms 'localization' $M_\mathfrak p$ of $M$ and the completion $M_\mathfrak p$ at $\mathfrak p$ ...
3
votes
1answer
30 views

valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
3
votes
3answers
172 views

Recommendations for Commutative Algebra Software?

I'd like a software that I can use to work with commutative algebra, specifically to figure out S-Polynomials, Buchberger's Algorithm, etc. I have Mathematica; if anyone could refer me to a package, ...
0
votes
0answers
49 views

Noetherian normal ring is a finite direct product of normal domains

Let $A$ be a Noetherian normal ring, that is, the localization of $A$ at every prime is a normal domain. I want to show $A$ is a finite product of normal domains. If $p_1,\ldots,p_n$ are the ...
0
votes
1answer
49 views

Why does a ring homomorphism induce a continuous map between spectra? [duplicate]

Let $\varphi: A \rightarrow B$ be a ring homomorphism. Let $f =\mathrm{Spec}(\varphi) : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ be the map associated to $\varphi$. Why is the map $f$ is continuous? ...
2
votes
0answers
20 views

Improvement of Buchberger's Algorithm (second part)

Suppose $S_j$ is a homogeneous syzygy of multidegree $\gamma_j$ in $S(G)$, where $G=\{g_1,\dots,g_t\}$. Show that $S_j G=\Sigma_{i=1}^{t} c_ix^{\alpha(i)}g_i$ has multidegree $< \gamma_j$. Now, I ...
1
vote
1answer
48 views

Question concerning the chinese remainder theorem for commutative rings

let $S$ be a commutative ring and $I_1,...,I_n\unlhd S$, such that $I_i+I_j=S\ \forall i\neq j$. Let $g_1,...,g_n\in S$. Why are there $h_1,...,h_n,h'\in S$, such that ...