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

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1answer
34 views

A submodule filtration that does not define the structure of topological module.

I was reading the following from Liu's Algebraic Geometry and Arithmetic Curves on page 18. In this book all rings are commutative and with unit. Let $A$ be a ring endowed with the $I$-adic topology. ...
15
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4answers
475 views

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$?

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$? Here tensor product is over the ring $\mathbb Z$ and $\mathbb Z[[X]] $ denotes formal power series over $\mathbb Z$. I think this ...
-1
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1answer
87 views

Canonical map is injective

Let $A$ and $B$ be commutative rings, and let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that for any $A$-module $M$, the canonical map $M\to M\otimes_AB$ is injective? I was ...
5
votes
1answer
94 views

Local cohomology killed by a power of I

Notations:: $H^i_I(M)$ is $i^{th}$ local cohomology of $M$ with support in $I$ and $H^i_I(M)=R^i\Gamma_I(M)$ where $R^i\Gamma_I(M)$ is the right derived functor of a covariant left exact functor, ...
3
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1answer
167 views

Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true? I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein. Question ...
0
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1answer
42 views

Cartesian product of projective system

Let $(M_i,\mu_i^j,I)$ and $(N_i,\nu_i^j,I)$ be two projective system of $R$-module ($R$ a commutative ring) How to prove that : $$\varprojlim_{i\in I}(M_i\times N_i)\cong \varprojlim_{i\in I}M_i\...
0
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1answer
37 views

Correspondence between valuations and valuation rings.

Matsumura gives us the following definition of an additive valuation. A map $\nu: K \rightarrow H \cup \{\infty\}$ from a field $K$ to $H \cup \{\infty\}$ is called an additive valuation or just a ...
2
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1answer
69 views

Inversion of fractional ideals with respect to localization

Let $R$ be a integral domain with field of fractions $K$, $S$ is any multiplicative set in R and $\mathfrak M$ is a fractional ideal of $R$. $$\mathfrak M^{-1}=\{x\in K:x\mathfrak M\subseteq{R}\}$$ ...
2
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1answer
61 views

Injective hull over different rings

Question: Does the injective hull of a module also depend on the ring?
5
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3answers
166 views

Module homomorphism question

I am trying to work on this problem: Let $R$ be a commutative ring, $I$ a nilpotent ideal of $R$, and $M,N$ $R$-modules. Let $\phi :M \to N$ be an $R$-module homomorphism. Show that if the induced ...
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1answer
68 views

Prove that the Zariski space $\text{Zar} \space (K,A)$ is compact.

I posted part of the proof from Matsumura's Commutative Ring Theory. I got stuck in the last sentence where it says "Hence the intersection of all the elements of $\mathcal{A}$ is the same thing as ...
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2answers
38 views

About radical of $(I,x)$ with $x$ irreducible

Let $I$ be a proper ideal of a polynomial ring $A$ and $x \in A$ an irreducible element. In a theorem of commutative algebra I will use the fact that, in this hypothesis, holds the following ...
2
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0answers
111 views

Local ring is an algebra over its residue field?

Suppose that that $X$ is an algebraic variety, and $V \subset X$ is an integral subvariety. In a paper I am trying to understand, it seems to be implicit that the local ring $\mathcal O_{X,V}$ (i.e. ...
2
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1answer
51 views

Different definitions of graded rings

In Atiyah i recently read the definition of a graded ring as a ring that can be written as $R=\displaystyle \bigoplus_{i \geq 0}^{\infty}R_i$ where each $R_i$ is an abelian subgroup of $R$ (with the ...
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3answers
44 views

nonNoetherian ring of countable cardinality

I recently asked myself if I could find a nonNoetherian ring (commutative w/ one) of countable cardinality. I could not. My wealth of nonNoetherian rings is small and usually relies on taking $k[...
2
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1answer
79 views

Non-isomorphic algebras with equal Hilbert-Poincaré series

Let $A,B$ be two finite-dimensional graded algebras and let $P_A(x),P_B(z)$ be theirs Poincaré series. Suppose now that $P_A(x)=P_B(z)$. Question. Is it implies that $A \cong B?$
3
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1answer
107 views

Why for every prime ideal there exists a submodule such that annihilator of submodule is this prime ideal?

Let $R$ be a commutative, not necessarily noetherian, ring with identity and $M$ a faithful and finitely generated $R$-module. For any prime ideal $P$ of $R$, there exists a prime $R$-submodule $K$ of ...
2
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1answer
117 views

$K[x,y]/\langle x^2-y^3\rangle \cong K[t^2,t^3]$ [duplicate]

I'm stuck with this (should be easy) computation. I started by considering the most natural map $K[x,y] \to K[t^2,t^3]$ which is the one that sends $x \mapsto t^3$ and $y \mapsto t^2$, and then ...
1
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1answer
369 views

if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian. Here is what I been told to do: Suppose that $R$ is not Noetherian, and use Zorn’...
2
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2answers
82 views

The finitely generated-ness of ideals $I +rR$ and $I:r$ imply $I$ is a finitely generated ideal [closed]

Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. Show that if the ideals $I +rR$ and $I:r=\{s∈R:sr∈I\}$ are finitely generated, then $I$ is a finitely generated ideal. Can anyone give ...
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1answer
31 views

prove that if $U\neq\mathbb{A}^n$ and $B_{(a,\epsilon)}\subseteq U$, then $U$ is not an affine algebraic set.

Without using hilbert's nullstellensatz prove that: i) $B_{(0,\epsilon)}$ it's not an affine algebraic set. (Where $B_{(a,b)}$ is the open disk with center $a$ and radius $b$) I prove this using ...
3
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0answers
45 views

Injectivity and localisation in Rings [duplicate]

Let $A, B$ be commutative rings with identity elements and let $\mathfrak{p} \subseteq B$ be a prime ideal. Let $\varphi: A \to B$ be an injective ring homomorphism. I want to show that the induced ...
2
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1answer
88 views

Why quotient ring of intersection of annihilators for a Jacobson ring is Jacobson?

Let $R$ be a commutative ring with identity and $M_i$ is a finitely generated $R$-module, for $i=1,\dots,n$. If for every $i$, $R/\operatorname{Ann}(M_i)$ is a Jacobson ring, why $R/\bigcap_{i=1}^n \...
2
votes
2answers
65 views

Integral extension is a finitely generated $R$-module?

Let $R$ be a commutative ring. If $b_1,\ldots,b_n$ are elements of a ring $R'$ (commutative) which are integral over $R$ then $R[b_1,\ldots,b_n]$ is a f.g. $R$-module. My question is: If $\{b_i\}_{...
0
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1answer
121 views

Calculating $Spec(\mathbb{C}[x]/\langle x^2 \rangle)$

I am currently trying to calculate $S = Spec(\mathbb{C}[x]/\langle x^2 \rangle)$. I'm pretty sure that $Spec(\mathbb{C}[x])$ is the set $\{x - \alpha : \alpha \in \mathbb{C} \} \cup \{0\}$ (ie - ...
3
votes
2answers
160 views

What is the injective hull of residue field of $R/m^t$?

Let $(R,m)$ be a local ring and $E=E(R/m)$ be the injective hull of $R/m$. Put $R_t=R/m^t$. What is the injective hull of residue field of $R_t$? I guess it is $(0:_E m^t) =\{x \in E \;:\;xm^t=0\}$....
3
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1answer
76 views

Cokernel of a faithfully flat homomorphism

Let $f:A\to B$ be faithfully flat ring homomorphism and $N=\operatorname{Coker}(f)$ the cokernel of $f$. Let $I$ be an ideal of $A$. How can I use the fact that if $B$ is a faithfully flat $A$-algebra,...
2
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2answers
84 views

Faithfully flat ring homomorphism properties

This is from Liu's Algebraic Geometry and Arithmetic Curves exercise 1.2.19 a. Let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that $f$ is injective and that $I\to I\otimes_AB$ ...
1
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1answer
36 views

Faitfully flatness over $B$ and flatness over $A$ equivalence

Let $B$ be an $A$-algebra, and let $E$ be a faithfully flat $B$-module. How can I show that $E$ is flat over $A$ if and only if $B$ is flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, ...
5
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0answers
86 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
3
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1answer
76 views

A finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$

How one can show that a finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.17.) I tried to ...
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0answers
71 views

Let $R$-algebra $A$. If $P⊂A$ is a minimal prime ideal then $p=P \cap R$ consists of zerodivisors for $A$?

We have: Let $R$ be a Noetherian commutative ring. Suppose $P⊂R$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. But how to prove this? The $R$-algebra $A$ is ...
1
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1answer
30 views

How to determine a primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$

I am trying to prove that the primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$, for a,b,c positive integers, is $(X^a,(X+Y+Z)^c) \cap (Y^b,(X+Y+Z)^c)$. The equality of the ideal and the ...
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0answers
34 views

No such prime ideal contains $I_1+I_2\implies I_1 $ and $I_2$ are relatively prime

It's clear to me that if $I_1$ and $I_2$ are two relatively prime ideals of a ring $R$, then there is no such prime ideal containing $I_1+I_2$, since by definition of relatively prime ideals $I_1+I_2=...
3
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2answers
145 views

Two questions about discrete valuation rings of varieties

Let $X$ be a proper, normal variety over $\mathbb{C}$, and $k(X)$ be its field of rational functions. I think the following two statements are true, but I was unable to give a proof or find the ...
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1answer
54 views

Prove that rings are isomorphic.

I read a text recently where it was explained how to compute the Hilbert function of $R=\mathbb{C}[x_0,...,x_n]$, as I was reading the author seemed to assume that for $f_1,...,f_i,f_{i+1} \in R$ we ...
2
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1answer
76 views

Krull dimension of $R[X]/(f(X))$ for $f(X)$ monic

How can I prove that the Krull dimension of $R[X]/(f(X))$, for $R$ a finitely generated noetherian integral domain and $f(X)$ monic, is equal to the Krull dimension of $R$? I don't even know where ...
3
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1answer
105 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} \...
2
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1answer
57 views

Is there a consensus on the correct way of raising elements of commutative rings to the power of $a/b$?

I'm trying to understand the "correct" way of raising elements of commutative rings to the power of $a/b,$ where $a$ and $b$ are integers, but not having much luck. Suppose $R$ is a commutative (...
2
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1answer
100 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. Shafarevich definition (pg 128) - A variety is non-singular in codimension one if the singular locus has codimension $> 1$. ...
0
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1answer
25 views

Show that $V(\bigcup_{i \in I} E_{i})=\bigcap_{i \in I} V(E_{i})$

This is a part of a problem in Atiyah's Introduction to Commutative Algebra introducing the Zariski Topology. Here we are given that $(E_{i})_{i \in I}$ is a family of subsets of a unital commutative ...
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0answers
84 views

Tensoring two short exact sequences

Let $R$ be a commutative ring with $1$ and consider the following short exact sequences of $R$-modules \begin{align} &0 \to M' \to M \stackrel{f}{\to} M'' {\to} 0 \qquad \text{and } \\ &0 \to ...
2
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1answer
109 views

Maximal ideal in local ring

The maximal ideal in $\mathbb{Z}_{(2)}$ should be $(2)$, but I don't understand this well. Suppose I take $\frac35\in \mathbb{Z}_{(2)}$. It is not in $(2)$ but in $(3).$ But what is the ideal between $...
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3answers
103 views

$\mathbb{Z}_{(2)}$ has one maximal ideal

My lecture notes state that the set $\mathbb{Z}_{(2)}$, defined as $$\mathbb{Z}_{(2)}:=\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{}\gcd(a,b)=1\text{ and } 2\nmid b\right\}$$ has a ...
1
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0answers
50 views

The prime ideals of the ring $K[x]$

I was wondering what the prime ideals of the ring $K[x]$ are, where $K$ is a ring. My guess is that it's any ideal generated by a set of irreducible polynomials over the ring $K$. Have I covered all ...
3
votes
1answer
89 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a $B$-...
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1answer
39 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
1
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1answer
20 views

Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
4
votes
2answers
156 views

What is the Krull dimension of $\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]$?

I am studying commutative algebra and saw the following question in one of the tests: What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$ I know ...
1
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1answer
70 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to ...