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

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2
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1answer
50 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
votes
1answer
76 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
105 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
votes
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
345 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 ...
1
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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
44 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
84 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
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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\}_{...
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1answer
120 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
156 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
75 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
votes
2answers
83 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 $I\to I\otimes_AB$ ...
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1answer
35 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, ...
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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
75 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
136 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
53 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
votes
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
votes
1answer
104 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
99 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
83 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
votes
1answer
108 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 $...
0
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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 ...
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0answers
46 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 $B$-...
0
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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 ...
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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
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2answers
151 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 ...
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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 ...
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1answer
148 views

What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an ...
2
votes
1answer
96 views

Quotient of ring is flat gives an identity of ideals

I have problem to understand and solve the exercise 1.2.14 on Qing Liu's book "Algebraic Geometry and Arithmetic Curves". It goes as follows: Let $A\to B$ be a ring homomorphism, and let $J$ be an ...
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0answers
94 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j \...
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0answers
68 views

Integrally closed domain.

Suppose $A$ is a unique factorization domain, $a$ is an element of $A$. Is the ring $A[x,a/x]$ always integrally closed? ($x$ is a variable over $A$) Thanks!
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1answer
84 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field $k$ with $\operatorname{char} k\not=2$, and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals $M$ ...
2
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1answer
198 views

Atiyah and Macdonald, exercise 11.7

I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian. The very beginning poses a problem, he states in the hint that: for a prime of height $m$ we can ...
3
votes
0answers
86 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions $\...
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1answer
77 views

how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
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votes
3answers
128 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that $f(X)=Z(\...
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0answers
32 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; \...
2
votes
1answer
72 views

Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
1
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1answer
112 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
3
votes
2answers
87 views

Flat algebra over a Dedekind domain

Let $B$ be a flat algebra over a Dedekind domain $A$. Let $f\in B$ be such that for every maximal ideal $\mathfrak m$ of $A$, the image of $f$ in $B/\mathfrak mB$ is not a zero divisor. How can I show ...