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

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2
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1answer
73 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 ...
10
votes
1answer
240 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
94 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
84 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
195 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
292 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 ...
0
votes
2answers
64 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 ...
2
votes
0answers
66 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 ...
0
votes
1answer
180 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
128 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 ...
3
votes
1answer
72 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 ...
9
votes
1answer
358 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 ...
2
votes
2answers
89 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 ...
9
votes
1answer
524 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 ring?
5
votes
1answer
121 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 ...
6
votes
2answers
181 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
vote
0answers
62 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 $R$?
4
votes
2answers
466 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
466 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
78 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 $$ ...
5
votes
0answers
82 views

approximating a variety locally by a vector space

Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$. Question 1: is there a way to approximate the variety ...
4
votes
1answer
86 views

Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
3
votes
1answer
80 views

Height of a prime ideal and number of generators of its localization

This question is very related to this one: generators of a prime ideal in a noetherian ring. Let $\mathfrak{p}$ be a prime ideal in a Noetherian ring and let $k$ be its height. Further suppose that ...
2
votes
1answer
86 views

Finite generation of Hom between cyclic and artinian module

Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $\operatorname{Hom}(R/I, M)$ finitely generated? Thanks.
2
votes
0answers
179 views

Question on integral scheme

Let $X=\operatorname{Spec}A$ be an affine scheme. In the book of Hartshone, he claimed that $X$ is integral if and only if $A$ is an integral domain. If $X$ is integral then we can deduce easily that ...
2
votes
1answer
241 views

Relation between spectrum of a ring and its quotient ring and localization.

Let $A$ be a commutative ring. $I$ be an ideal of $A$, $S$ be a multiplicative closed subset. We know that : there is 1-1 correspondence between the prime ideals $\mathfrak{p}\in Spec A$ containing ...
7
votes
2answers
182 views

Property of modules via exact sequences

Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence $$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} ...
5
votes
1answer
277 views

Number of generators of the maximal ideals in polynomial rings over a field

Hi I'm trying to prove the following If $K$ is a field (not necessary algebraically closed) then every maximal ideal of $K[x_{1},\dots,x_{n}]$ is generated by exactly $n$ elements. I know that ...
5
votes
1answer
292 views

A subset of a field that is a subfield

It can be verified that the following assertion is true: a subset $S$ of a field $F$ is a subfield if $S$ contains the additive and multiplicative identities 0 and 1, if $S$ is closed under addition, ...
5
votes
1answer
129 views

Can we have a Primary Avoidance Theorem ?

Prime Avoidance Theorem says: Let $ P_1, P_2,\dots, P_n $ be prime ideals in a commutative ring $R$ and let $I$ be an ideal of $R$ such that $ I \subseteq P_1 \cup P_2 \cup \cdots \cup P_n$. ...
2
votes
1answer
66 views

Maximal element in set of Ann(m) for m in M is prime

Exercise 15.1.32 in Dummit & Foote, along with the included hint, is Suppose that $M$ is a $R$-module and that $P$ is a maximal element in the collection of ideals of the form ...
5
votes
2answers
126 views

Change of variables in $k$-algebras

Suppose $k$ is an algebraically closed field, and let $I$ be a proper ideal of $k[x_1, \dots, x_n]$. Does there exist an ideal $J \subseteq (x_1, \dots, x_n)$ such that $k[x_1, \dots, x_n]/I \cong ...
3
votes
1answer
55 views

Question about completions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. I have some questions about Corollary 10.3 and Corollary 10.4. Why the sequence $$ 0 \to \frac{G'}{G' \cap G_n} ...
1
vote
1answer
155 views

Questions about the intersection of all neighborhoods of $0$ in a topological abelian group.

Let $H$ be the intersection of all neighborhoods of $0$ in a topological abelian group. On page 102 of the book introduction to commutative algebra by Atiyah and Macdonald, the fourth line of the ...
3
votes
1answer
92 views

Atiyah-Macdonald, Proposition 2.12, uniqueness of the tensor product.

The following is a result from Atiyah-Macdonald, defining and showing existence and uniqueness of tensor product of modules over a commutative ring. Proposition 2.12. Let $M, N$ be $A$-modules. ...
2
votes
0answers
43 views

Relation between inverse limits (and direct limits) with limits in calculus. [duplicate]

What is the relation between inverse limit (and direct limit) with limits in calculus? Are there some special cases that an inverse limit (or direct limit) is a limit in calculus (for example, the ...
1
vote
0answers
156 views

Integral extension inside a polynomial ring over a field

Let $K$ be a field and $D = K[X]$. I need to show that if $f\in D$ is non constant, then the extension of rings $K[f]\subset D$ is integral, and if $A$ is a subring of $D$ which contains $K$ and has ...
3
votes
0answers
115 views

associated graded ring is the quotient of a free algebra by a homogeneous ideal

Let $A$ be a semilocal ring with Jacobson radical $m$ and let $I$ be an ideal of definition, i.e. an ideal such that $m^{\nu} \subset I \subset m$. Consider the associated graded ring of $A$, given by ...
1
vote
1answer
93 views

maximal ideal properly contains union of its square with the union of minimal prime ideals

One of the first theorems one encounters in the study of commutative algebra is that if $I$ is an ideal of a ring $A$ not contained in any of the prime ideals $P_1,\cdots,P_n$, then $I$ is not ...
2
votes
1answer
48 views

Question about inverse limits.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On Page 104, I have some questions about the proof that $\{A_n\}$ is surjective implies $d^A$ is surjective. We have ...
4
votes
1answer
79 views

Question about homomorphisms $f_{!}, f^{!}$.

Let $f: A \to B$ be a finite ring homomorphism and $N$ a $B$-module. $N$ can be considered as an $A$-module if we define $A \times N \to N$, $(a, n) \mapsto f(a)n$. Therefore we have a map $f_{!}: ...
3
votes
1answer
83 views

Question about the lying over theorem.

I have some questions about the proof of the Lying over theorem in the book Introduction to commutative algebra by Atiyah and Macdonald. (1) In the proof of Theorem 5.10 of Page 62, is the map ...
5
votes
1answer
71 views

Radical of prime ideal in homogeneous localization is prime

Let $B$ be a graded ring, $B=\oplus_{d\ge 0} B_d$. If $f\in B$ is homogeneous, we let $B_{(f)}$ denote the subring of $B_f$ made up of elements of the form $af^{-N}$, $N>0$, where $a$ is a ...
0
votes
1answer
95 views

Are these prime ideals?

Let $R=\mathbb Z[\sqrt{-5}]$. I want to show $P=3\,R+(1+\sqrt{-5})\,R$ and $Q= 3\,R+(1-\sqrt{-5})\,R$ are prime ideals of $R$.
3
votes
1answer
132 views

Radical of ideals in local one dimensional rings

Let $R$ be a local one dimensional ring. I want to show that for all $ a,b\in R$, $\sqrt{Ra+Rb}$ is equal to $\sqrt{Ry}$ for some $y\in Ra+Rb$ or is equal to $R$.
2
votes
1answer
124 views

Questions about Grothendieck groups.

I have a question of the exercise 26 on page 88 of the book introduction to commutative algebra by Atiyah and Macdonald. In 26(iii), let $A$ be a field. Then finitely generated $A$-modules are finite ...
0
votes
1answer
102 views

The relation between minimal prime ideals and nilpotents

Show that a prime ideal $I$ of a ring $R$ is minimal if and only if for each $x\in I$ there exists $a\in R\setminus I$ such that $ax$ is nilpotent.
0
votes
1answer
102 views

Question about zero-divisors and a quotient of a polynomial ring by an ideal in the book Introduction to commutative algebra by Atiyah and Macdonald.

I am reading the book the book Introduction to commutative algebra by Atiyah and Macdonald. I have two questions On Page 51. On Line 5 of Page 51, it is said that the zero-divisors in ...
0
votes
1answer
35 views

Question about primary decompositions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On page 50, Line -7, it is said that "if $f: A \to B$ and $\mathfrak{q}$ is a primary ideal in $B$, then ...
3
votes
0answers
124 views

If $M \otimes M \simeq M$ is there anything we can say about $M$? [duplicate]

Over a commutative (and unital) ring, if $M \otimes M \simeq M$ can we say anything about $M$? If we base change to a point, ie tensor with a map from the ring into a field, then $M$ becomes a vector ...