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

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0
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1answer
151 views

The clique number of zero-divisor graphs

If $R$ is finite commutative ring with exactly $8$ elements, show that the clique number of the zero-divisor graph is $2$. Edit. Let $R$ be a commutative ring and $Z(R)$ be the set of all zero-...
2
votes
1answer
70 views

In $\mathbb{C}[x]$ is it true that $F_{a,b}=\{p\in\mathbb{C}[x] : p(a)=p(b)\}$ for $a\neq b$ is a maximal subring?

The problem is in the title. It is clear that $F_{a,b}$ is a ring, but it is not so clear to me that it is maximal in $\mathbb{C}[x]$. I tried to consider it as a vector space and show that it has ...
-1
votes
1answer
136 views

Show that if an ideal is free as a module then it is principal.

Here $\mathfrak a$ is an ideal of a commutative ring $A$. Show that $\mathfrak a$ is principal if it is free as an $A$-module.
14
votes
5answers
1k views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
2
votes
0answers
85 views

Infinitely many primary decompositions of an ideal

Given a noetherian ring $R$ and an ideal $I$, we know that if the associated primes of $I$ coincide with its minimal primes (i.e. $\text{Min}(I)=Ass(I)$ ) , then there is a unique irredundant primary ...
2
votes
0answers
134 views

(Finite) continued fractions over a general domain

I am looking for some literature (articles or books) where finite continued fractions over a general integral domains (that is, in a fraction field of that domain, but the "coefficients" are from the ...
2
votes
3answers
168 views

Confusion about Spec of quotient ring

Consider the ring $A := \dfrac{\mathbb C[x]}{(x(x-1)(x-2))}$. According to some sources (cf. Vakil) $sp(Spec (A))$ should be just the three points $\{0,1,2\}.$ It seems right, because $A$ is the ring ...
3
votes
1answer
139 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} \...
6
votes
2answers
451 views

Nilradical of absolutely flat ring

Suppose $A$ is an absolutely flat ring (i.e. every $A$-module is flat). Is it true that nilradical of $A$ is trivial, i.e. $\mathfrak{N}(A)=\{0\}$? I believe the answer is yes. Here is my attempted ...
1
vote
1answer
164 views

Determining generators for vanishing ideal of projective closure

This question is along the lines of 2.9 from Hartshorne. Notation: Let $S:=k[x_0,\ldots,x_n]$ be the coordinate ring of $\mathbb{P}^n$, and let $A:=k[y_1,\ldots,y_n]$ be the coordinate ring of $\...
3
votes
1answer
159 views

An affine open neighborhood of a nonsingular point

Let $X$ be an algebraic variety over an algebraically closed field $k$. Here a variety is an integral separated scheme of finite type over $k$ as in Hartshorne's book. Let $x \in X$ be a closed point. ...
3
votes
1answer
138 views

degree of the Hilbert polynomial of a quotient

Let $A=\bigoplus_{n \ge0} A_n$ be a Noetherian graded ring with $A_0$ Artinian. Suppose that $A=A_0[a_1,\dotsc,a_d]$ with $a_i$ having degree $1$. Let $M$ be a finitely-generated graded $A$-module. ...
3
votes
2answers
221 views

Does product distribute with respect to intersection for ideals in a ring.

Let $I,\, J$ and $K$ be ideals in a commutative ring $R$. Could you please give an example such that $(I\cap J)K = IK\cap JK$ is not true?
2
votes
0answers
81 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
4
votes
1answer
119 views

Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
1
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1answer
36 views

Help with this definition of $(G:_M I)$

I didn't understand why in this definition $I$ has to be an ideal to make sense. REMARK This is from Steps in Commutative Algebra, page 107. Thanks a lot
6
votes
1answer
288 views

Finite set of zero-divisors implies finite ring

Show that any commutative ring $R$ having only $n$ non-zero zero divisors ($n\geq 1$) is finite and doesn't contains more than $(n+1)^2$ elements.
0
votes
0answers
47 views

Rees algebra of a monomial ideal [duplicate]

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that $$ R[It]/\...
-1
votes
1answer
74 views

Intersection of radical primal ideal

Let $A$ a noetherian ring, $a_{1},...,a_{n}$ primary ideals, with $rad(a_{i})=m_{i} $ maximal ideal and $m_{i}\neq m_{j}$ si $i\neq j$. How can I prove that $a_{1}\cap a_{2}\cap \ldots\cap a_{n}=a_{1}\...
7
votes
0answers
473 views

Regular Noetherian local rings are integral domains - questions about the proof

I am reading a proof that if $(A,\mathfrak m)$ is a regular local ring, then $A$ is an integral domain. I put the major questions I'm worried about in bold, but there are a lot of little things I'm ...
5
votes
2answers
353 views

Commutative rings whose non-trivial ideals are maximal

It is well known that a local ring is a ring containing only one maximal ideal. I was wondering if there is a characterization (or any information) of the commutative rings such that all their non-...
1
vote
1answer
115 views

Question on normal Noetherian local rings

Consider a normal Noetherian local ring $(A,\mathfrak m)$ of dimension $1$. I am working through a proof that such a ring is a principal ideal domain. Consider $x\in \mathfrak m \backslash \mathfrak ...
1
vote
1answer
82 views

Normal at every localization implies normal

I'm having some trouble with basic ring theory. Let $A$ be an integral domain and $\alpha$ an element of its fraction field integral over $A$. I am trying to understand a proof that $\alpha\in A$ ...
1
vote
0answers
42 views

Factoring maps between noetherian rings

Let $A,B$ be commutative noetherian rings, and let $f:A\to B$ be a ring map. Can one always factor $f$ as $A\to C\to B$ where $C$ is a noetherian ring, $A\to C$ is flat, and $C \to B$ is surjective?
1
vote
1answer
245 views

Integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$

I am trying to compute the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i].$ I have managed to show that $\mathbb{Z}[i]$ is inside the integral closure, and I suspect it is the entire thing. Can ...
2
votes
0answers
47 views

Homocyclic primary module over PID

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following ...
2
votes
1answer
148 views

Finitely Generated Modules over Quotient of a DVR.

Let $(R,t)$ be a DVR with uniformizing parameter $t$ and let $M$ be a finitely generated module over $R/(t^n)$. Then $M\cong \bigoplus_r R/(t^r)$. Question: How many summands of type $R/(t^r)$, where ...
2
votes
1answer
218 views

Property of an edge ideal

User fbakhshi deleted the following question: Let $G$ be a simple graph with finite vertex set $X = \{x_1,\dots, x_n\}$. The edge ideal of $G$, denoted by $I = I(G)$, is the ideal of $R=K[x_1,\dots,...
4
votes
1answer
215 views

Rees algebra of a monomial ideal

User fbakhshi deleted the following question: Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\...
1
vote
2answers
39 views

$k \subset A \subset B$, $B\supset k$ f.g., $\text{codim}_k(A) < \infty$ $\Rightarrow$ $B \supset A$ f.g. module?

Does this hold? Let $k \subset A \subset B$ where $k$ is a field and $A,B$ are commutative rings. If $B$ is a finitely-generated ring over $k$ and $\dim_k(B/A) < \infty$ then $B$ is a finitely-...
2
votes
1answer
80 views

disputing a length equality in Matsumura (fundamental theorem of dimension theory)

Let $A$ be a semilocal Noetherian ring with Jacobson radical $m$ and $M$ a finite $A$-module. Let $x \in m$. According to Matsumura's Commutative Ring Theory p. 99 (Step 2), $l(xM/xM\cap m^n M)=l(M/(m^...
10
votes
1answer
254 views

Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?

I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the "...
1
vote
1answer
114 views

intuition in definition of the dimension of a finite module

Let $A$ be a commutative ring and $M$ a finitely generated $A$-module. Then the dimension of $M$ is defined to be the dimension of the quotient $A/ann(M)$, where $ann(M)$ stands for the annihilator of ...
3
votes
3answers
104 views

Prime ideal in the ring of polynomials

I'm trying to do the following: Let $R = K[X,Y,Z]$ and $\mathfrak{p}$ = $(X+Y,Z^{2}-X)$. Show that $\mathfrak{p}$ is prime and find the transcendence degree of $R/\mathfrak{p}$. If I prove ...
8
votes
1answer
259 views

Construction(s) of new integral domains from “old ones”

Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...
6
votes
2answers
647 views

Maximal ideals in rings of polynomials [duplicate]

Let $k$ be a field and $D = k[X_1, . . . , X_n]$ the polynomial ring in $n$ variables over $k$. Show that: a) Every maximal ideal of $D$ is generated by $n$ elements. b) If $R$ is ring and $\...
1
vote
2answers
79 views

The action of a Galois group on a prime ideal of a Dedekind domain

This is a slight variant of a question I asked earlier. Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $...
3
votes
0answers
119 views

The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If $\frak{P}...
3
votes
1answer
459 views

Graded commutative $R$-algebras

Let $R$ be a commutative ring and $T$ a graded commutative $R$-algebra. This means that $\,T$ consists of a collection $\{T_n\}_{n\geq 0}$ of $\,R$-algebras, where the elements of $R_n$ are called ...
3
votes
1answer
155 views

Krull dimension of a $\mathbb Q$-algebra

I'm trying to find the Krull dimension of $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$. My professor said that I have to consider that $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$ is a $\mathbb{Q}$-algebra but I don'...
3
votes
1answer
89 views

showing that the power series in two indeterminates over a field has dimension 2

Let $k$ be field and consider the power series $A=k[[x,y]]$. What is the simplest way (in the sense of using the least "heavy" theorems) to show that $\operatorname{dim} A=2$, where $\operatorname{dim}...
10
votes
1answer
678 views

A question on faithfully flat extension

This question arose while reading page 116 of Red Book by Mumford. Let $B$ be a faithfully flat extension of $A$. Can I claim that $b \otimes 1 = 1 \otimes b$ in $B\otimes_A B$ if and only if $b\...
2
votes
2answers
94 views

The relation between the intersection of two prime ideals and the annihilator of two elements of them

Let $P,Q$ be two prime ideals such that $P\cap Q\neq{0}$. Let $a\in P\setminus Q$ and $b\in Q\setminus P$ such that $ab\neq0.$ Show that if $P\cap Q\subseteq \text{Ann}(a)\cup \text{Ann}(b)$, then $P\...
11
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1answer
1k views

Every maximal ideal is principal. Is $R$ principal?

Let $R$ be a commutative ring with 1. If every maximal ideal of $R$ is principal, is $R$ a principal ideal ring?
5
votes
1answer
174 views

Valuation ring in an algebraically closed field

Let $k$ be an algebraically closed field and $(R, \mathfrak{m})$ a valuation ring in $k$ i.e. the field of fractions of $R$ is $k$. Then Mumford (Red Book, page 127) claimed that the residue field $R/\...
6
votes
2answers
299 views

Hilbert polynomial of disjoint union of lines in $\Bbb{P}^3$

Let $X$ be the disjoint union of the two lines in $\Bbb{P}^3$ given by $Z(x,y)$ and $Z(z,w)$. Letting $R = k[x,y,z,w]$, I have computed the following free resolution for the homogeneous coordinate ...
1
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0answers
79 views

Is an invertible ideal in a semi-quasilocal ring a principal ideal?

Let $R$ be a semi-quasilocal ring and $I$ be an invertible ideal of $R$. Is $I$ a principal ideal of $R$?
8
votes
2answers
1k views

Principal ideal and free module

Let $R$ be a commutative ring and $I$ be an ideal of $R$. Is it true that $I$ is a principal ideal if and only if $I$ is a free $R$-module?
6
votes
2answers
1k views

Show that a ring with disconnected spectrum is a product of two subrings. [duplicate]

It's an exercise from the book introduction to commutative algebra by Atiyah and Macdonald. If $\operatorname{Spec}(A)$ is disconnected, I'm asked to show that $A$ is a product of two subrings. I ...
1
vote
1answer
96 views

Localization and extension of rings

Is $\mathbb{Z}_{(3)}[i,\sqrt{2}]=(\mathbb{Z}[i,\sqrt{2}])_{(3)}$ (where by subscript $(3)$ we mean localization at the ideal generated by $3$)? Do both of these rings contain elements like $$ \frac{...