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

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4
votes
1answer
80 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
1
vote
1answer
93 views

relation between units and non zero divisors in a ring

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?
3
votes
2answers
71 views

Can $ℂ$ be viewed as a (nontrivial) field of fractions?

Is there an interesting ring $S ⊂ ℂ$ such that $ℂ = Q(S)$? I’m thinking no, but how can I prove it?
1
vote
0answers
52 views

Proving that a certain local ring is regular

I understand that this is a special case of the Jacobian criterion, but I was hoping that there was a simpler argument to prove it than for the criterion itself (I don't fully understand the proof of ...
4
votes
1answer
93 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ==============...
3
votes
1answer
87 views

Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, as ...
1
vote
1answer
155 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
8
votes
3answers
303 views

Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement (...
2
votes
1answer
120 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
2
votes
1answer
63 views

prove that this ideal is radical

Let $A=\mathbb k[x,y,z]$ and let the ideal $$ I=(z-1,x^2-y).$$ I need to find $rad(I)$ but i don't know how. I think that this ideal is radical but I don't know good criteria for doing that =(
1
vote
1answer
58 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
2
votes
1answer
45 views

Ideals agreeing in a localization

I have an integral scheme $X$, and two coherent ideal sheaves $\mathcal I$ and $\mathcal J$ on $X$. I know there is a (maybe not closed) point $x$ of $X$ such that $\mathcal I$ and $\mathcal J$ ...
6
votes
1answer
117 views

The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local ...
4
votes
0answers
118 views

Flatness and Cohen-Macaulay rings

Let $A$ be a local Artin ring, $R$ a local Noetherian ring, $f:A \to R$ a flat morphism and $R$ is cohen-Macaulay. Let $I$ be an ideal in $R$ such that $R/I$ is also Cohen-Macaulay. Under what ...
2
votes
0answers
68 views

Proof of the Jacobian criterion - book of Eisenbud

I could really use some help understanding a statement in the last part of the proof of the Jacobian criterion in "Commutative Algebra with a view toward Algebraic Geometry" by D. Eisenbud, namely: ...
3
votes
2answers
66 views

Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= \text{...
2
votes
1answer
191 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
128 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
134 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
240 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 $Q'=ax²+bxy+...
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
152 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
68 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
198 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 is $...
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
62 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
147 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
122 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
111 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 out....
4
votes
3answers
1k 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
90 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
145 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
68 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
67 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
61 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 ...
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vote
0answers
55 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
128 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 $k[x,...
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
135 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
74 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
102 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
153 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
47 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 ...