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

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3
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1answer
85 views

Associated graded ring of a Fermat cubic

Let $R$ be a graded Fermat cubic, i.e. $R$ is a graded ring given by $$ R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3), $$ with a standard grading $\operatorname{deg}(x)= ...
3
votes
1answer
150 views

Non-isomorphic $\mathbb{C}$-algebras

The question is as follows: Show that the $\mathbb{C}$-algebras: $A=\mathbb{C}[x,y]/(x^2y-xy)$, $B=\mathbb{C}[x,y]/(x^2y+xy^2)$, $C=\mathbb{C}[x,y,z]/(xy, yz, zx)$, and ...
2
votes
1answer
69 views

Basic constructions for graded algebras.

I'm reading about the Weil algebra of a Lie group and it involves some constructions I'm not very familiar with, for instance the "free graded-commutative graded algebra on $a_1...a_n$ with degrees ...
3
votes
2answers
137 views

Maximal and Prime Ideal

Consider a field $K$. I have seen that the ideal $(Y-X^2,Z-X^3)$ is a prime ideal in $K[X,Y,Z]$. But now I have to see that it is not a maximal ideal. Therefore, I have to find an ideal between ...
0
votes
1answer
52 views

A variety $X\subset\mathbb{A}^n$ has two disjoint components iff $A(X)$ is the product of 2 finitely generated reduced k-algebras

It is a homework: Show that a variety $X\subset\mathbb{A}^n$ has two disjoint components if and only if the coordinate ring $A(X)$ may be written as the product of two finitely generated ...
2
votes
1answer
78 views

Identity between roots of polynomials

Let $A\in{\mathbb C}[X]$ be a monic polynomial of degree $n\geq 2$, with roots $\alpha_1,\alpha_2,\alpha_3, \ldots ,\alpha_n$. Let $B$ be the polynomial $$ B=\prod_{k=1}^{n} ...
0
votes
1answer
77 views

That submodule generated by one element leads to submodule being finitely generated

In Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, in the prove of Proposition 1.4, the auther seems to use the following fact. Let $R$ be a Noetherian ring, $M$ is a ...
4
votes
1answer
106 views

Is the direct limit of Noetherian rings necessarily Noetherian?

Is the direct limit of Noetherian rings necessarily Noetherian? And if it is, how to prove; if it is not,what is a counterexample. (I was thinking this question: if $A_{m}$ are Noetherian for $m\in ...
1
vote
1answer
115 views

Krull dimension of this local ring

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
0
votes
1answer
127 views

What is the Krull dimension of this local ring

I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$. I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
1
vote
2answers
84 views

Question on decomposing a noetherian ring into product of PIDs

Let $R$ be a reduced noetherian rings of dimension $d$, $S$ be a multiplicative set of all regular elements of $R$, and $K=S^{-1}R$ be localisation of $R$. Show that $K[T]$ (polynomial in 1 ...
4
votes
1answer
117 views

What is an example of a Dedekind ring that is non-principal?

Prop. 15 of Serge Lang's ANT shows that a sufficient condition for a Dedekind ring $R$ to be principle is that it only have finitely many primes. To give an outline of the argument, one starts with a ...
4
votes
1answer
54 views

Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
5
votes
0answers
81 views

“Graded free” is stronger than “graded and free”

This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a ...
0
votes
0answers
149 views

Subring with maximal ideals (prime avoidance). Proof verifying and small question

Let $t∈\Bbb N$ and let $p_1, \dots ,p_t$ be $t$ distinct prime numbers. Show that $$R = \{α∈\Bbb Q : α = m/n \mbox{ for some } m ∈ \Bbb Z \mbox{ and } n∈\Bbb N \mbox{ such that } n \mbox{ is ...
0
votes
0answers
49 views

The quotientf of a $k$-algebra $A$ by $k$?

Let $A$ be a $k$-algebra, for some field $k$ of $\mathop{\mathrm{char}}0$ (I believe it can be a ring of arbitrary character... but anyways) then am I wrong? $A/k$ is just the past of $A$ "without ...
3
votes
3answers
192 views

Example of invertible maximal ideal that is not generated by one element

Could anyone give me an example of an invertible maximal ideal of some integral domain which is not generated by one element?
2
votes
1answer
179 views

Prove that in a Noetherian ring, no invertible maximal ideal properly contains a nonzero prime ideal

Let $R$ be an integral domain which is Noetherian, let $P$ be an invertible maximal ideal, and let $Q<P$ be a prime ideal. How to show that $Q=0$? I have proved that $Q=QP$, and still haven't used ...
1
vote
0answers
39 views

Showing that a ring $R$ is Noetherian [duplicate]

The problem is as follows: Let $R$ be a ring such that for each maximal ideal $m$ in $R$, the localization $R_m$ is noetherian and each non-zero element of $R$ is contained in only finitely many ...
5
votes
0answers
99 views

Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following: (1) If $J_k$, $k=1,\dots,n$, are ...
2
votes
1answer
115 views

Factor ring by a regular ideal of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is a finitely generated $A$-module. It is well-known that ...
2
votes
1answer
62 views

on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

if $A$is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$. Also on ...
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vote
2answers
100 views

Quotient of maximal ideals by its power giving simple modules

So I understand that $R/m$ where $m$ is a maximal ideal would give simple module. My question is, would $m/m^2$ also give a simple module? My progress thus far: My first approach was to realize that ...
6
votes
1answer
227 views

Existence of minimal non-zero prime ideals: Counter examples?

Let $R$ be an integral commutative ring with unit. If $R$ is noetherian, then every ideal has finite height, in particular, there exist minimal non-zero prime ideals if (and only if) $R$ is not a ...
2
votes
1answer
47 views

Find $\mathbb{C}[x^2,x^3]\cap \mathbb{C}[(x-1)^2, (x-1)^3]$.

Find $\mathbb{C}[x^2,x^3]\cap \mathbb{C}[(x-1)^2, (x-1)^3]$. I am trying to find the above subring. I would prefer hints more than complete solutions.
-1
votes
1answer
71 views

Hilbert Basis Theorem applied to integral domains

Was reading a solution to an exercise of the Atiyah-MacDonald "Introduction to commutative Algebra" and this passage catched my attention "Let $A$ be an integral domain, so by H-B-T we can infer ...
2
votes
0answers
77 views

Roots of polynomials with coefficients in an algebraically closed field

I have encountered to a famous problem about polynomials. Could you please show me the outline of achieving to it. Question: Let $\kappa$ be an algebraically closed field. How could we show that ...
0
votes
0answers
73 views

Lying over “from above”

Let $B/A$ be a integral extension of (commutative unital) rings. The "Lying over" theorem states that for any prime ideal $P$ in $A$ there is a prime ideal $Q$ in $B$ such that $Q\cap A=P$. The usual ...
2
votes
2answers
57 views

Isomorphism of $\mathcal{O}_K$-modules

I came across the following claim in K Conrad's notes: Let $L/K$ be a finite extension of number fields, For fractional ideals $\mathfrak{a}, \mathfrak{b}$, and $\mathfrak{c}$ of $\mathcal{O}_L$, with ...
2
votes
2answers
165 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
2
votes
1answer
177 views

Composition Series of $R$-modules

Recently I have been reading about composition series and lengths of modules. Everything that I have encountered has been rather sparse on examples. In this vein, I was wondering how you would compute ...
1
vote
1answer
55 views

Is Noetherianity necessary for quasi-regularity to imply regularity?

Theorem 16.3 In Matsumura (CRT) reads as follows: Let $A$ be a Noetherian ring, $M \neq 0$ an $A$-module, and $a_1,\dots,a_n \in A$; set $I=(a_1,\dots,a_n)A$. Under the condition (*) each of ...
0
votes
1answer
96 views

Inconsistent system of simultaneous equations

Let $F$ be an algebraically closed field, and $f_1,\ldots,f_n$ polynomials in $k$ variables over $F$. The system of simultaneous equations $$\mathcal{F}: ...
1
vote
1answer
133 views

Valuation Ring with value group $\Gamma=\mathbb{Z} \oplus \mathbb{Z}$

Let $\Gamma=\mathbb{Z}\oplus \mathbb{Z}$ be the free abelian group with two generators, lexicographically ordered. That is, $(a,b)\geq (a^{'},b^{'})$ iff either $a>a^{'}$ or $a=a^{'} \mathrm{and}\, ...
2
votes
2answers
1k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
1
vote
1answer
50 views

About minimal prime ideals and varieties

Let $W$ be a variety, and $I=\mathbb{I}(M)$, then we have $$ I=\operatorname{rad}(I)=P_1\cap\cdots\cap P_n $$ where $P_i$'s are minimal prime ideals containing $I$. Thus we have $$ ...
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votes
1answer
62 views

Discrete valuation rings are infinite.

Assume $F\supset k$ is a functional field. Assume $R\subset F$ is a discrete valuation ring with a quotient filed $F$, that contains the field of constants $k$. Assume $t$ is a local parameter for ...
0
votes
1answer
87 views

Discrete valuations of a functional field have discrete valuation rings.

Theorem: If $\nu:F\to\mathbb R\cup\{\infty\}$ is a valuation of a functional field, then the set $$\mathfrak O_{\nu}=\{x\in F: \nu(x)\geq 0\}$$ is a local ring with maximal ideal $$\mathfrak ...
0
votes
3answers
86 views

Sufficient condition for a ring to be a product of two rings [duplicate]

In his algebraic geometry notes, Vakil suggests the exercise (remark 3.6.3) of showing that a ring $A$ is a product $A = A_1 \times A_2$ iff $\operatorname{Spec} A$ is disconnected. His hint is to ...
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votes
3answers
271 views

Tensor product and injective maps

during a class we met a map $\Phi : \mathbb{Z}^n\to \mathbb{Z}^n$. We saw that $\Phi\otimes 1_\mathbb{Q}$ was an isomorphism and then the Professor said that it implied that $\Phi$ was injective. ...
0
votes
1answer
246 views

This kernel is trivial?

Let $M_0,\ldots,M_N$ be all the monomials of degree $d$ in the $n+1$ variables $x_0,\ldots,x_n$, where $N=C_{n+d,n}-1$. Let $\theta:k[y_0,\ldots,y_N]\to k[x_0,\ldots,x_n]$ be the homomorphism ...
2
votes
0answers
75 views

Help in this proposition of Hartshorne's book

I'm trying to understand this proposition in Hartshorne's book: The theorem 1.8A says I put $B=K[X_1,\ldots, X_n]$, am I right? If so, why $K[X_1\ldots,X_n]/(f)=Z(f)$? Thanks
0
votes
1answer
34 views

on the extension module of a pair (some module, a finite free module)

Let $A$ be a commutative ring and $M$ an $A$-module that admits a projective resolution. According to my understanding it is true that $\operatorname{Ext}^i(M,A^n) \cong ...
2
votes
0answers
53 views

The integral closure of Bezout domains in arbitrary field extensions is Prüfer?

Let $R$ be a Bezout domain with quotient field $K$, $L$ an arbitrary extension field of $K$, and $\overline{R}$ the integral closure of $R$ in $L$. Is $\overline{R}$ a Prüfer domain? If the answer is ...
1
vote
1answer
53 views

Proof that $\bigcap_{n=1}^\infty J^n=0$ in commutative noetherian ring

If we let $R$ be a commutative noetherian ring. Then $\bigcap_{n=1}^\infty J^n=0$ where $J$ is the jacobson radical of $R$ Proof. Denote $X=\bigcap_{n=1}^\infty J^n=0$. Then let $XJ=Q_1\cap \ldots ...
2
votes
1answer
47 views

How to calculate the derivative of a function in $\mathbb{Q}_p$?

On Wikipedia it is stated that the function $$ f:\mathbb{Q}_p\to \mathbb{Q}_p $$ with $f(x)=(1/|x|_p)^2$ if $x\neq 0$ and $f(0)=0$ is differentiable and its derivative is the zero-function. How ...
1
vote
1answer
56 views

Characterization of an invertible module

Let $B$ be a commutative ring. Let $A$ be a subring of $B$. If $M$ and $N$ are $\mathbb{Z}$-submodules of $B$, we denote by $MN$ the submodule of $B$ generated by the subset $\{ab\mid a \in M, b\in ...
0
votes
1answer
45 views

Basic Issue With the Hom Functor on Commutative Rings

In the category of $A$-modules, one has the following property: if $f:M \rightarrow M''$ is a map of $A$-modules, and the induced map $f^*:Hom(M'',N) \rightarrow Hom(M,N)$ is injective for all $N$, ...
2
votes
1answer
81 views

Question on the formal completion of a ring $R$ w.r.t. an ideal $J$

I am trying to understand the completion $\hat R$ of a commutative unitary ring $R$ w.r.t. an ideal $J$. Please let me first recall, what I think is true (since if there is a misunderstanding already, ...
0
votes
1answer
40 views

Why does the maximal irrelevant is out of this correspondence?

I'm solving the Hartshorne's questions and I didn't understand why $S_+$ doesn't occur in this equivalence: My reasoning By the previous exercise, if $X$ is an algebraic set in $\mathbb P^n$, we ...