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

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2
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2answers
50 views

Polynomial algebra and polynomial ring

What is the difference between polynomial algebra and polynomial ring? because sometimes I read polynomial algebra and it looks like a polynomial ring $K[x,y,..]$ in many variables. Thanks
0
votes
0answers
29 views

Having only the zero as a nilpotent element is a local property

I want to show that having only the zero as a nilpotent element is a local property for a Ring $R$. Assume $R$ only has the zero element as a nilpotent element and there exists a prime ideal $p$ ...
0
votes
1answer
31 views

An example of a c.i./Gorenstein/C.M. integral domain which is not integrally closed

If I am not wrong, it is known that: {Regular rings} $\subsetneq$ {Complete intersection rings} $\subsetneq$ {Gorenstein rings} $\subsetneq$ {Cohen-Macaulay rings}. It is known that a regular ring ...
6
votes
2answers
135 views

Does there exist such an invertible matrix?

Let $n \geq 1$ and $A = \mathbb{k}[x]$, where $\mathbb{k}$ is a field. Let $a_1, \dots, a_n \in A$ be such that $$Aa_1 + \dots + Aa_n = A.$$ Does there exist an invertible matrix $\|r_{ij}\| \in ...
0
votes
2answers
108 views

Artinian rings have finite length

In a recent question of mine here I asked whether it is true or not that Artinian (commutative) rings have finite length. I came up with a proof, and I want to know if it is valid. So, I want to ...
0
votes
0answers
42 views

How to understand this sentence within Atiyah-Macdonald's textbook about commutative algebra

In page 102 of this textbook, authors mentioned that: Assume topological group $G$ has a fundamental system of neighborhoods consisting of subgroups as: $G= G_0 \supseteq G_1 \supseteq\cdots\supseteq ...
0
votes
2answers
54 views

Why the dimension of $R/(a)$ is $0$?

How do I see the following fact? If $R$ has dimension $1$, and $a$ is a non-zerodivisor and non-unit, then $R/(a)$ has dimension $0$. That is saying if $P_1\supset P_2\supset (a)$ are two prime ...
0
votes
1answer
45 views

What's wrong with the following argument that every module is flat?

Okay, I know I'm doing something incredibly stupid here, but for whatever reason I can't figure out what. As I understand it, an R-module M is flat iff $f : I \otimes_R M \to I M$ is an isomorphism ...
2
votes
1answer
52 views

If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
1
vote
0answers
45 views

dimension formula for fiber product of affine varieties

Let $X \subset \mathbb{A}^n, \, Y \subset \mathbb{A}^m, \, Z \subset \mathbb{A}^{\ell}$ be irreducible affine varieties and let $f: X \rightarrow Z, \, g: Y \rightarrow Z$ be surjective morphisms. ...
0
votes
1answer
28 views

Given $A$-modules $N \subset M$ such that $N_m=M_m$ for all maximal ideals $m$, show that $M=N$

I am working on this exam question 6 $A$ is commutative ring with $1$ a) If $N \subset M$ are $A$-modules and $N_m=M_m$ for all maximal ideals $m$, show that $M=N$. We know that $N_m=M_m$ ...
7
votes
0answers
75 views

What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
0
votes
0answers
27 views

Projective ideal of a Non-noetherian domain

If $R$ is an integral domain which is not Noetherian and let $I$ be an ideal which is not finitely generated. We have always, if I is invertible, then I is always projective. Is the converse true when ...
3
votes
2answers
56 views

Flat Non Projective $A$-Module [duplicate]

A standard fact in Commutative Algebra is that a Projective $A$-module is flat. The converse is false. Can someone show me an example of a Flat Non Projective $A$-Module? Thank you!
3
votes
1answer
37 views

Redundancy in the definition of Dedekind domain?

Is there a domain which is noetherian and whose nonzero prime ideals are maximal, but which is not integrally closed? This may be a silly question to experts. I ask because I think I have found ...
0
votes
1answer
38 views

Isomorphic Affine Schemes

If $f:A\to B$ is a homomorphism of rings such that $f':\text{spec} B \to \text{spec} A$ is a homeomorphism does it follows that the spectra are isomorphic as schemes? I was able to reduce this ...
3
votes
1answer
44 views

Local ring and isomorphism problem

I have a local ring $R$ with maximal ideal $\mathfrak{m}$. Fixing some $x\in\mathfrak{m}$, I want to show that $\mathfrak{m}^{k-1} \subset (\mathfrak{m}^k : x)$ and conclude that $R/(\mathfrak{m}^k : ...
0
votes
1answer
29 views

Basic problem about local ring and dimension

Probably this is just a misunderstanding, but it's making me waste a lot of my time. I'm studying commutative algebra through Kemper's book. There is the following statement in chapter 12 (dimension ...
3
votes
1answer
46 views

vanishing ideal of product of two affine varieties

Let $X \subset \mathbb{A}^n$, $Y \subset \mathbb{A}^m$ be affine varieties and let us consider them embedded in disjoint subspaces of $\mathbb{A}^{n+m}$. Let $p \in k[x_1,\dots,x_n,y_1,\dots,y_m]$ be ...
2
votes
2answers
59 views

Example for an ideal which is not flat (and explicit witness for this fact)

I'm looking for an ideal $\mathfrak{a}$ of an commutative (possibly nice) ring $A$ together with an injective $A$-module homomorphism $M\hookrightarrow N$ such that the induced map ...
3
votes
1answer
27 views

$M/x_nM$ finitely generated over $ k[x_1,…, x_{n-1}] $ as graded module

I'm trying to figure out the following: Let $M $ be a finitely generated graded module over $S=k[x_1,..., x_n]$ with standard monomial grading. Let $K $ be the kernel of multiplication by $x_n$ in ...
6
votes
2answers
144 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
2
votes
1answer
43 views

Minimal free resolution of the twisted cubic

This is exercise 13.15 in Harris' book "A First Course...". Let $X$ be the twisted cubic with ideal $I(X) = (XZ-Y^2,YW-Z^2,XZ-YW).$ Let $S(X)$ denote the homogeneous coordinate ring of $X$ and ...
1
vote
1answer
38 views

Atiyah-MacDonald: proof of Proposition 7.9, weak Nullstellensatz.

Proposition 7.9 in Atiyah & MacDonald's Introduction to Commutative Algebra states: Let $k$ be a field and $E$ a finitely-generated $k$ algebra. If $E$ is a field, then it is finite algebraic ...
2
votes
1answer
45 views

Dimension of local ring by a zero-divisor

Let $(A,m)$ be a Noetherian local ring and let $x \in m$. If $x$ is a non-zero divisor, then we know that the Krull dimension satisfies $\dim A/(x) = \dim A-1$, see e.g. Atiyah-MacDonald Corollary ...
3
votes
1answer
74 views

AG on non-Noetherian rings

I must apologize beforehand as this question is pretty basic, but I can't seem to find a satisfying answer in the introduction section of the book I'm currently reading (if there is a page on here, I ...
1
vote
1answer
18 views

Maximal ideal of valuation

Let $\nu : K \rightarrow G \cup \{\infty\}$ be a map defined by, where $G$ is a totally ordered group and $g < \infty$ for all $g\in G.$ $\nu(a) = \infty$ if and only if $a = 0,$ $\nu(a + b) ...
19
votes
1answer
287 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
4
votes
0answers
60 views

Proposition 11.3 in Atiyah MacDonald

Let $A=\bigoplus_{n=0}^{\infty} A_n$ be a Noetherian graded ring, in which case $A$ is generated as an algebra over $A_0$ by elements $x_1,\dots,x_s$ of degrees $k_1,\dots,k_s$. Let $\lambda$ be an ...
12
votes
0answers
103 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
1
vote
1answer
36 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
3
votes
2answers
38 views

Flat extension of noetherian rings and formal power series

Let $A \to B$ be a flat homomorphism of Noetherian rings. Is it true that it induces a flat homomorphism of formal power series $A[[x]] \to B[[x]]$?
23
votes
1answer
452 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Do we have a precise description of $\mathfrak A$? Is ...
1
vote
1answer
27 views

Two dimensional valuation domain

Let $R$ be a two-dimensional valuation domain with prime ideals $0 \subset P \subset M$ and value group $G=\Bbb Z \oplus \Bbb Q$. Then $M^2=M$ and $P^2 \neq P$. Why $M^2=M$ and $P^2\neq P$? Can we ...
0
votes
1answer
47 views

Element invertible in integral extension of ring implies invertible in ring [duplicate]

Please excuse some minor hiccups in terminology, I am primarily reading this in Swedish so feel free to correct any. Let $A\subseteq B$ be an integral extension and $\alpha\in A$ an invertible ...
3
votes
2answers
60 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
1
vote
1answer
25 views

Find the integral closure of an integral domain in its field of fractions [duplicate]

Let $k$ be a field and let $R = k[x,y]/(x^2-y^2+y^3)$. Note that $R$ is an integral domain. Let $F$ be the field of fractions of $R$. How to determine the integral closure of $R$ in $F$? I have ...
1
vote
1answer
18 views

Radical of the annihilator of an element of a Noetherian module

Assume $M$ is a commutative Noetherian $R$-module and $m\in M$ is such that $P=\sqrt{\operatorname{Ann}(m)}$ is a prime ideal in $R$. Is it true that $P$ is an associated prime of $M$, i.e. there is ...
0
votes
1answer
123 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
2
votes
1answer
80 views

$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's Advanced Topics in The Arithmetic of Elliptic Curves (whose proof is left to the reader): Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
0
votes
1answer
29 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be a $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
2
votes
0answers
38 views

Generalization of the Going up Theorem to arbitrary chains of prime ideals

Let $S$ and $R$ be commutative rings with $1$. This is the usual form of the Going up theorem that one encounters in commutative algebra texts: Let $S$ be integral over $R$, and suppose that we have ...
1
vote
1answer
55 views

Theorem 31.7 of Matsumura, Commutative Ring Theory

Theorem: If A is a Noetherian local ring and A[x] catenary, then A is formally catenary. In the proof, it is assumed that A is integral domain and A* (the completion of A) is not equidimensional and ...
8
votes
1answer
80 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
2
votes
0answers
55 views

When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
0
votes
0answers
55 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
2
votes
2answers
56 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
2
votes
1answer
42 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
0
votes
0answers
52 views

Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...
5
votes
1answer
145 views

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...