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

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2
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1answer
167 views

Is the localization of a ring $R$ at a prime ideal a finitely generated algebra over $R$?

Let $R$ be a ring and let $S=\{1,s,s^2,s^3,\dots\}$ be a multiplicative system of $R$. Consider the canonical map $R\rightarrow S^{-1}R$. Is $S^{-1}R$ a finitely generated algebra over $R$? It looks ...
4
votes
1answer
125 views

Atiyah-MacDonald Ch. 4 exercise 20: what's the module analogue of $\sqrt{\mathfrak{a}+\mathfrak{b}} = \sqrt{\sqrt{\mathfrak{a}}+\sqrt{\mathfrak{b}}}$?

Atiyah-MacDonald exercises 20-23 in chapter 4 develop a theory of primary decomposition for modules, in analogy with the theory developed in the chapter for rings. Exercise 20 begins with this ...
5
votes
1answer
130 views

Castelnuovo-Mumford regularity of a Veronese subring

I've faced a problem while reading a paper. It is mentioned to be trivial but I couldn't prove it. I'd appreciate if you can lead me to some resources or if you can prove it for me. Thank you. ...
1
vote
2answers
74 views

Are any of these rings isomorphic?

As part of my ongoing struggle to understand the complex conics, I've reached the following problem: Let $Q_1 = x^2 + y^2$, $Q_2 = x^2 - 1$, and $Q_3 = x^2$ be polynomials in $\mathbb{C}[x,y]$. ...
2
votes
1answer
238 views

Can a multiplicatively closed subset contain zero?

Let $A$ be a ring and $S$ be a multiplicatively closed subset. Can $S$ contain $0$? If so what will happen if we do $S^{-1}A$? A concrete and easy example coming to my mind is $A = \Bbb Z$, and $S = ...
-1
votes
0answers
60 views

Classes of rings C[x,y]/(x²+cy²+ey+f) [duplicate]

I have a question. I would like to describe the classes of rings that appear in $\mathbb{C}[x,y]/I$ up to isomorphism, where $I=(Q)$, $Q=x²+cy²+ey+f$, $c,e,f\in\mathbb{C}$. $Q$ comes from ...
2
votes
1answer
52 views

Describing integral closure of quadratic number fields

I'm facing the following problem. Let $p$ be a prime and $ K=\mathbb{Q}(\sqrt{p}) $. I'm trying to find the integral closure of $ \mathbb{Z} $ in $ K $. I don't really know where to start. I've ...
2
votes
1answer
26 views

Proof with exact sequence of modules

I'm trying to prove that if the sequence $$ M \xrightarrow{\varphi} W \rightarrow 0$$ is exact with $ W $ being a free module, then $ M \simeq \ker{\varphi} \oplus W $ What I got is that since $ W ...
3
votes
1answer
139 views

A ring with ACC on prime ideals, whose spectrum is non-noetherian.

I am currently working on the converse of the exercise #12 on chapter 6 of Atiyah-Macdonald's book on commutative algebra. The problem is asking whether there is a ring $A$, which satisfies the ...
3
votes
0answers
67 views

Homology of Derivations of a dgca algebra

Let $(A,d)$ be a differential graded commutative and associative algebra. A derivation on $A$ is a linear endomorphism $L: A \to A$, that satsfies $L(ab)= L(a)b+ aL(b)$. More general a derivation of ...
5
votes
1answer
180 views

Normalisation of $k[x,y]/(y^2-x^2(x-1))$

I am trying to figure out the normalisation of $k[x,y]/(y^2-x^2(x-1))$, for an algebraically closed field $k$. I can show that it is not normal and I have the information that the normalisation ...
1
vote
1answer
66 views

Is $k[x^4,x^3y,xy^3,y^4]$ a local ring?

I noticed that a system of parameters is defined in local rings and some books say that $\{x^4,y^4\}$ is a system of parameters for $R=k[x^4,x^3y,xy^3,y^4]$. Is $R$ a local ring or we use it refers to ...
1
vote
1answer
61 views

Good book for Local Fields/ Commutative algebra?

I am currently studying Local Fields from Serre's textbook, but finding that it requires a bit too much prior knowledge for me. Can anyone suggest another book that I can use alongside Serre that ...
1
vote
1answer
142 views

non-examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of ...
1
vote
1answer
116 views

Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}[x,y]$

This is a continuation of the question I asked here. The problem is now: Let $Q = ax^2 + bxy + cy^2 + dx + ey + f \in \mathbb{C}[x,y]$ be a general quadratic polynomial, that is, $a,b,c \not= 0$. ...
1
vote
1answer
105 views

Trouble showing flatness

Let $K$ be a field and $\pi: K[x]/(x^2) \to K$ be the ring homomorphism given by the valuation at $0$. I'm stuck in showing that $\pi^*(K)$ (the pullback) is not a flat module (over $K[x]/(x^2)$).
1
vote
1answer
64 views

Can anyone help me understand an application of Nakayama lemma?

In the Wikipedia there is an application of Nakayama lemma: In the special case of a finitely generated module $M$ over a local ring $R$ with maximal ideal $m$, the quotient $M/mM$ is a vector ...
0
votes
1answer
35 views

Prove that some local noetherian integral domain is a field

A local noetherian integral domain $A$ is a field if the unique maximal ideal $m$ satisfies $m^n = m^{n+1}$ for some $n\in N$ I think it should be related to Nakayama lemma, but cannot figure it ...
2
votes
3answers
906 views

Commutative artinian ring is noetherian

Suppose R is a commutative Artinian ring then R is Noetherian. I am aware of the proof which uses the idea of filtration. But I would like to prove this fact without that idea but haven't got far ...
2
votes
1answer
89 views

A problem with tensor products

Let $K$ be a field, $R=K[x^2,x^3]$, $S=K[x]$, and consider $S$ as an $R$-module. Given $f: S \to R \oplus R$ so that $f:p \mapsto (x^3p,-x^2p)$, prove that $f\otimes 1: S \otimes_R S \to (R\oplus R) ...
0
votes
1answer
130 views

Understanding the proof (via primary decomposition) the “ideal factorization theorem” in Dedekind domains

I am trying to understand the outline of the strategy for proving (via primary decomposition) that every non-zero ideal of a Dedekind domain can be expressed uniquely (up to the order of the factors) ...
0
votes
0answers
66 views

Is this ideal a prime ideal?

Let $k$ be a field and $k[x_1,x_2,x_3,x_4]$ a polynomial ring in four variables over $k$. How can we show that the ideal $(x_3^3-x_2^2x_4, x_4^3-x_1^2x_3, x_3x_4-x_1x_2, x_2x_4^2-x_1x_3^2)$ is prime? ...
1
vote
1answer
66 views

Confusion about the definition of localization

For example, let $\mathbb Z$ be the ring and $S = \mathbb Z - 2\mathbb Z$. Then the quotient ring should be: $S^{-1}A = \{a/s: a\in \mathbb Z \text{ and }s \in S\}$, which is formed by the equivalence ...
1
vote
2answers
57 views

Image of the map induced on spectra

Apologize in advance if this is a bit trivial but I am stuck on the following: Prove that for $\varphi : R \to S$ a map between commutative rings, the prime $\mathfrak{p}$ is in the image of the ...
1
vote
0answers
50 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
1
vote
1answer
90 views

Prime ideals in $A$ and prime ideals in $S^{-1}A$

Let $A$ be a ring and $S$ be a multiplicative closed subset. Then there is a 1 to 1 correspondence between the prime ideals in $A$ (intersect $S$ is empty) and prime ideals in $S^{-1}A$. My question ...
2
votes
3answers
126 views

$k[t^{a_1},t^{a_2},t^{a_3}]$ in the form $k[x,y,z]/(…)$

I want to write $k[t^6,t^7,t^{15}]$ in the form $k[x,y,z]/(...)$; but I even don't know how to start. Is there in general a way that one can write $k[t^{a_1},t^{a_2},t^{a_3}]$ in the form ...
0
votes
1answer
44 views

construction of $S^{-1}A$

If $A$ is a ring and $S$ a multiplicative set, how does the elements of $S^{-1}A$ look like? In my book one introduces the equivalence relation $\sim$ on $A \times S$ as follows: $(a,s) \sim (b,t) ...
2
votes
4answers
133 views

Counterexample in Dedekind domains

Let $K$ be a number field and $\mathcal O_K$ the ring of algebraic integers in $K$. If $\mathfrak p$ is a prime ideal, then $\mathcal O_K/\mathfrak p$ is finite field. My question is: Finding a ...
-2
votes
2answers
71 views

$A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring

Question: Suppose $A$ is a commutative ring, $P$ is a prime ideal. Prove $A_P$ is local ring. I have no idea how to construct the unique maximal ideal.
0
votes
1answer
92 views

Associated non-minimal prime ideal

I am trying to find an example of a noetherian local ring with an associated prime of height greater or equal 1. That is, I want a noetherian local ring $R$ together with an associated prime $p$ ...
1
vote
0answers
145 views

Ambiguity in the definition of unmixed ideal

Compare the definitions: Page 136 Matsumura, Commutative ring theory: A proper ideal $I$ in a Noetherian ring $A$ is said to be unmixed if the heights of its prime divisors are all equal. ...
2
votes
1answer
46 views

Is an ideal prime when its complex extension is prime?

Let $I = \langle f_1,\dots,f_k\rangle$ be an ideal in $\mathbb R[x_1,\dots,x_n]$. The same $f_i$ generate an ideal $\widetilde I$ in $\mathbb C[x_1,\dots,x_n]$. When $\widetilde I$ is prime in ...
1
vote
0answers
47 views

Flatness on the affine line for a coherent sheaf

Let $A:=\mathbb{C}[t], M$ a finitely generated $A$ module. Denote by $m_\alpha$ the maximal ideal generated by $t-\alpha$ for $\alpha \in \mathbb{C}$, $S_\alpha$ the multiplicative set which is the ...
2
votes
1answer
91 views

Extension of R-linear derivation to localization

Let $R$ be a commutative ring. Given a commutative $R$-algebra $A$, a multiplicative subset $S \subset A$, and a $R$-linear derivation $D: A \rightarrow M$, where $M$ is an $S^{-1}A$-module, $D$ can ...
0
votes
1answer
33 views

$\overline{V(I)-V(J)}=V(\bigcup_{n=1}^{\infty}I\colon J^n)$

Is it true that $\overline{V(I)-V(J)}=V\left(\bigcup_{n=1}^\infty I\colon J^n\right)$? If not, is it true for noetherian rings?
1
vote
1answer
99 views

Show that $Rad(I)$ is a prime ideal

The ring $R$ is commutative with unit. An ideal $I$ is called primary, if it stands the following: If $ab \in I$ then $a \in I$ or $b^n \in I$, for a natural number $n$. Show that if $I$ is a ...
4
votes
1answer
139 views

Does any (noetherian) integral domain have a “UFD closure”?

Let $R$ be a (possibly noetherian if that helps) commutative unital integral domain. Does there exist a UFD $\overline{R}$ such that $R$ embeds in $\overline{R}$ (via some map $\psi$) and such that ...
1
vote
2answers
202 views

Irreducible components of affine variety

Fix some algebraically closed field $k$ and let $X$ be the affine variety given by the ideal $I=(z^2-xy,xz-z)$, how can I describe the irreducible components of $I$? I know that there is a bijection ...
1
vote
2answers
54 views

Localization Question: $\frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)}$

Questions: $\rm\color{#c00}{(1)}$ Is the $[\Longrightarrow]$ implication of $$ \frac{a}{b}\in\left(\mathbb{Z}_{(p)}\right)^{\times}\iff\frac{b}{a}\in\mathbb{Z}_{(p)} $$ obvious? ...
3
votes
2answers
498 views

Prime ideals in an arbitrary direct product of rings

By ring I mean commutative unital ring. The prime ideal structure of a finite direct product of rings is well known: For $\prod_{i=1}^n R_i$, it is of the form $\prod_{i=1}^n P_i$ where only one ...
6
votes
0answers
172 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
2
votes
0answers
88 views

Computing a valuation of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are transcendental over $k$. I want to compute the valuation ring of $F$, the field of fractions of the ring $A=k[x,y]/I$, where $I=\langle ...
1
vote
1answer
103 views

Question on Generic Freeness, ref. [Matsumura, page 185]

I am sure this must have been answered somewhere but I can't find them, so I shall try my luck here. Let $A$ be a Noetherian integral domain and $M$ a finitely generated $A$-module. Then there ...
1
vote
1answer
117 views

Discrete Valuation Rings - Atiyah & MacDonald

The following is claimed (without much proof) during the the proof of Prop 9.2 in Atiyah & MacDonald. Saurabh commented below giving the proof that was probably intended by A&M (thank you!). I ...
0
votes
1answer
227 views

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain?

When is the quotient ring of a multivariable polynomial ring over a field by a monomial ideal an integral domain? I am actually trying to show that a monomial ideal is prime by showing the ...
3
votes
0answers
86 views

Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, ...
0
votes
1answer
89 views

Generic flatness on modules

I am looking for a stronger notion of generic flatness. Let $A$ be a Noetherian ring, $M$ a finitely generated module over $A$. Suppose there exists a maximal ideal $m$ of $A$ such that $M_m$ (the ...
4
votes
2answers
184 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no prime ideal $Q$ such that $0 \subsetneq Q \subsetneq P$. ...
0
votes
1answer
131 views

Artin local ring [duplicate]

I have studied "structure theorem for Artin rings" which states "An Artin ring $A$ is unique a finite direct product of Artin local rings". Let $A$ be Artin ring. By Chinese remainder theorem, $A ...