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

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16
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1answer
451 views

Modules with $m \otimes n = n \otimes m$

Let $R$ be a commutative ring. Which $R$-modules $M$ have the property that the symmetry map $$M \otimes_R M \to M \otimes_R M, ~m \otimes n \mapsto n \otimes m$$ equals the identity? In other ...
15
votes
3answers
535 views

Can a prime in a Dedekind domain be contained in the union of the other prime ideals?

Suppose $R$ is a Dedekind domain with a infinite number of prime ideals. Let $P$ be one of the nonzero prime ideals, and let $U$ be the union of all the other prime ideals except $P$. Is it possible ...
15
votes
4answers
760 views

Why is ideal more important than subring?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
15
votes
1answer
1k views

Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
15
votes
2answers
1k views

A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
15
votes
1answer
541 views

When does the modular law apply to ideals in a commutative ring

Let $R$ be a commutative ring with identity and $I,J,K$ be ideals of $R$. If $I\supseteq J$ or $I\supseteq K$, we have the following modular law $$ I\cap (J+K)=I\cap J + I\cap K$$ I was wondering ...
15
votes
2answers
208 views

If the tensor product of two modules is free of finite rank, then the modules are finitely generated and projective

If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ are finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
15
votes
1answer
191 views

Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
15
votes
3answers
491 views

*writing* proofs involving commutative diagrams

This question is a little fuzzy so might be closed, but I'll give it a shot. I'm sorry this question has quite a long introduction, I don't see how to formulate it more concisely. In modern algebraic ...
15
votes
1answer
276 views

A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
15
votes
1answer
155 views

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
14
votes
5answers
1k views

A non-noetherian ring with $\text{Spec}(R)$ noetherian

Question 1: Does such a ring can be found? Note: The definition of a noetherian topological space is similar to that in rings or sets. Every descending chain of closed subsets stops after a finite ...
14
votes
3answers
459 views

Question about UFD

I want to know some examples with the following properies. Let $R$ be a domain such that every non unit element $x$ is a product of finite irreducible elements,but $R$ is not a UFD, and there is ...
14
votes
3answers
1k views

Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
14
votes
1answer
255 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
14
votes
2answers
1k views

What does a zero tensor product imply?

I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any cmmutative ring), the tensor product $M\otimes_A N$ is zero iff $\operatorname{Ann}(M)+\operatorname{Ann}(N)=A$. ...
14
votes
4answers
466 views

Isomorphism between quotient rings of $K[X,Y]$

Let $K$ be a field of characteristic $0$ and $m,n\in\mathbb Z$, $m,n\ge 1$. Prove that $$K[X,Y]/(X^2-Y^m)\simeq K[X,Y]/(X^2-Y^n)$$ if and only if $m=n$. (Related to Isomorphism between quotient rings ...
14
votes
2answers
175 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
13
votes
5answers
2k views

In a principal ideal ring, is every nonzero prime ideal maximal? [duplicate]

Inspired by this question, I was wondering whether from just the hypothesis that $A[X]$ is a nontrivial (commutative) principal ideal ring (so without supposing it is a domain) one can deduce that $A$ ...
13
votes
2answers
537 views

Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
13
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2answers
442 views

Usefulness of completion in commutative algebra

After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions: (i) What is the usefulness of the concept of completion in ...
13
votes
5answers
644 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
13
votes
2answers
303 views

Example of two prime ideals whose intersection of the squares not equal to the square of the intersection

In this topic the OP raised the following question: Let $R$ be a commutative noetherian ring and $\mathfrak p,\mathfrak q \in \operatorname{Spec}(R)$. Is it true that $(\mathfrak p\cap \mathfrak ...
13
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2answers
566 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
13
votes
3answers
682 views

Localization at a prime ideal is a reduced ring

Here is the question that I came up with, which I am having trouble proving or disproving: Let $A$ be a ring (commutative). Let $p \in Spec(A)$ such that $A_p$ is reduced. Then there exists an open ...
13
votes
1answer
356 views

History of Commutative Algebra

There are books of the history of Algebraic Geometry, there are also papers about it (All had done by J.Dieudonné). But I could not find any book or paper about the history of Commutative Algebra. ...
13
votes
5answers
692 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 ...
13
votes
1answer
300 views

$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic

Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
13
votes
1answer
410 views

Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
13
votes
1answer
515 views

Exercise 2.17(d) of Eisenbud's Commutative Algebra

First some notation: Let $P$ be a homogeneous prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a ...
13
votes
1answer
140 views

Modules over Completion

I have the following question. Let $R$ be a commutative ring with unit, and let $\hat{R}$ denote its completion (w.r.t. any ideal $I$). Let $M$ be an $\hat{R}$-module. Is $M= N\otimes_R \hat{R}$ for ...
12
votes
3answers
586 views

What are rational integer coefficients?

I have a question about the following excerpt from Atiyah-Macdonald (page 30): “A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means ...
12
votes
4answers
2k views

Surjective endomorphisms of finitely generated modules are isomorphisms

My Problem: Let $M$ be a finitely generated $A$-module and $T$ an endomorphism. I want to show that if $T$ is surjective then it is invertible. My attempt: Let $m_1,...,m_n$ be the generators of ...
12
votes
2answers
392 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
12
votes
2answers
636 views

Fields finitely generated as $\mathbb Z$-algebras are finite?

Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra. (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite?
12
votes
2answers
343 views

Integral domain with fraction field equal to $\mathbb{R}$

I wonder if there is an integral domain $A\subseteq \mathbb{R}$ which is not a field, and such that the field of fractions of $A$ is equal to $\mathbb{R}$? Edit: here as a possible direction: it is ...
12
votes
2answers
549 views

Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
12
votes
2answers
1k views

Inverse Image of Maximal Ideals

Given a map of commutative rings with unit, it is often the case that the inverse image of a maximal ideal is not maximal. For example, consider the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$. ...
12
votes
5answers
856 views

Finite quotient ring of $\mathbb Z[X]$

Since userxxxxx (I don't remember the numbers) deleted his own question which I find interesting, let me repost it: Let $f,g\in\mathbb Z[X]$ with $\mathrm{gcd}(f,g)=1$. Prove that the ring ...
12
votes
1answer
455 views

Regular local ring and a prime ideal generated by a regular sequence up to radical

Let $R$ be a regular local ring of dimension $n$ and let $P$ be a height $i$ prime ideal of $R$, where $1< i\leq n-1$. Can we find elements $x_1,\dots,x_i$ such that $P$ is the only minimal prime ...
12
votes
1answer
157 views

What's the motivation of the definition of primary ideals?

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$ Primary ideals can be regard as the generalization of prime ideals and radical. But ...
12
votes
1answer
201 views

An inverse limit

Let $k$ be a field. Consider the inverse limit $\varprojlim k[x,y]/(y\cdot x^n)$. I wonder if there is a nice description of this ring? Geometrically, we look at the union of the line $y=0$ ...
12
votes
2answers
148 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
12
votes
1answer
159 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
12
votes
1answer
195 views

Characteristic of a finite ring with $34$ units

Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the ...
12
votes
2answers
393 views

The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...
12
votes
1answer
371 views

How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?

Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can ...
12
votes
2answers
782 views

Homomorphisms of graded modules

Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ ...
12
votes
0answers
395 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
11
votes
4answers
5k views

Example of modules that are projective but not free; torsion-free but not free

Free modules are projective, and projective modules are direct summand of free modules. Is there any example of projective modules that are not free? (I know this is not possible for modules of ...