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

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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 ...
3
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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 ...
1
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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 ...
2
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2answers
175 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
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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 ...
3
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69 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 ...
0
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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
46 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. ...
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1answer
51 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
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2answers
56 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
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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
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0answers
59 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
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1answer
96 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
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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. ...
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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 $M$ ...
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1answer
53 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 ...
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3answers
191 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?) ...
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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 ...
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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 ...
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1answer
24 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
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0answers
16 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
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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 ...
9
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0answers
165 views

Trace and determinant over a commutative ring

Copy pasted from Darij Grinberg's post on AoPS from a while back: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=63&t=276930 Let $R$ be a commutative ring with unity. For any matrix ...
0
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1answer
40 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 ...
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2answers
85 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 $r\neq\dim{R}$? ...
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1answer
43 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
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1answer
70 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
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1answer
32 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 ...
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Some problems with Castelnuovo-Mumford regularity

I have some problems to complete my thesis. In this paper: "Upper bound for the Castelnuovo-Mumford regularity of associated graded modules" - Cao Huy Linh Lemma 4.1 Please explain for me about ...
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98 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 ...
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2answers
94 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
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0answers
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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
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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, ...
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1answer
71 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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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
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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
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3answers
162 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, ...
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1answer
47 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 ...
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1answer
46 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? ...
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0answers
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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 ...
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1answer
44 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 ...
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Radical of an ideal in a finitely generated ring over $k$ is the intersection of maximal ideals containing it. [duplicate]

From Matsumura p.34 Let $k$ be a field, $A$ a ring which is finitely generated over $k$, and $I$ a proper ideal of $A$; then the radical of $I$ is the intersection of all maximal ideals containing ...
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Improvement of Buchberger's Algorithm

1) Let $S(F)$ be the subset of $(k[x_1,\dots,x_n])^s$ consisting of all syzygies on the leading terms of $F=\{f_1,\dots,f_s\}$. Then every element of $S(F)$ can be written uniquely as a sum of ...
2
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1answer
29 views

2 questions concerning identities of closed subspaces of $spec(S)$ for a commutative ring $S$

I have the following questions: Let $S$ be a commutative ring and let $M,N$ be closed subspaces of $spec(S)$, such that $M\cap N=\emptyset$. 1) Why are there ideals $I_1,I_2\unlhd S$, such that ...
2
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1answer
42 views

integral ring homomorphism

Consider a homomorphism $f: A\to B$ of commutative rings and let $b\in B$. Let $g\colon A[X]\to B[X]$ be defined by $g(X) = X$. Put $I = g^{-1}((bX-1))$ (contraction of the ideal $(bX-1)\subseteq ...
0
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0answers
40 views

Reference request: Cartier divisors versus invertible sheaves by Kleiman

Please delete this question if it is deemed inappropriate. Could someone link me to the paper "Cartier divisors versus invertible sheaves" by Kleiman please? My library doesn't provide access to it. ...
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0answers
17 views

Inverse image of a maximal ideal under a morphism of finitely generated $\mathbb{C}$-algebras. [duplicate]

Let $$ f: A\to B $$ be a morphism of finitely generated $\mathbb{C}$-algebras, suppose $\mathfrak{m}\unlhd B$ is a maximal ideal, I want to show that $f^{-1}(\mathfrak{m})$ is a maximal ideal of $A$. ...
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1answer
69 views

Finding a coordinate ring

I am having hard time in calculating (or constructing) $\displaystyle\frac{\mathbb C[x,y]}{\langle y^2 - x^3 - x\rangle}$. I tried homogenizing the ideal $y^2 - x^3 -x $ to $ wy^2 - x^3 - xw^2$. But ...