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

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19
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2answers
774 views

Is the radical of an irreducible ideal irreducible?

Fix a commutative ring $R$. Recall that an ideal $I$ of $R$ is irreducible if $I = J_1 \cap J_2$ for ideals $J_1$ and $J_2$ only when either $I = J_1$ or $I = J_2$. Question : Assume that $I$ is an ...
19
votes
3answers
379 views

Bound on nilpotency index of endomorphisms

Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we ...
18
votes
2answers
933 views

Did Zariski really define the Zariski topology on the prime spectrum of a ring?

The question is not: “Did Zariski really define the Zariski topology?” It is: “Did Zariski really define the Zariski topology on the prime spectrum of a ring?” Here is the motivation. --- On page ...
18
votes
2answers
479 views

Basic counterexample re: preimages of ideals

I'm trying to think of an example of a homomorphism of commutative rings $f:A\rightarrow B$ and ideals $I,J$ of $B$ such that $f^{-1}(I)+f^{-1}(J)$ is not a preimage of any ideal of $B$. I can't seem ...
18
votes
1answer
379 views

Is $\mathbb{Z}$ the only totally-ordered PID that is “special”?

(All my rings are commutative and unital.) Definition. Call a totally-ordered ring $R$ special iff for all non-zero $b \in R,$ every coset of $bR$ has a unique element in the interval $[0,|b|).$ ...
18
votes
1answer
476 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 ...
18
votes
0answers
404 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
17
votes
3answers
2k views

$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$

I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold. This is exercise 2.11 ...
17
votes
2answers
2k 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 ...
17
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2answers
822 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
17
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3answers
2k 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 ...
17
votes
1answer
539 views

An exercise with Zariski topology

I read this exercise: Prove that the set $S = \{ (n, 2^n, 3^n ) \mid n \in \mathbb{N} \}$ is dense in $\mathbb{C}^3$ with Zariski topology. I have seriously thought about it, but I do not manage to ...
17
votes
2answers
2k 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 $\operatorname{Ann}(M)+\operatorname{Ann}(N)=A$. ...
17
votes
1answer
1k views

Does localisation commute with Hom for finitely-generated modules?

Question. Let $R$ be a ring, $\mathfrak{p}$ a prime, $M$ a finitely-generated $R$-module, and $N$ any $R$-module. Is the natural map $$\textrm{Hom}_R(M, N)_\mathfrak{p} \to ...
17
votes
1answer
255 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 ...
17
votes
1answer
430 views

The polynomial whose roots are all real

Suppose $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb{R}[x]$ is a polynomial whose roots are all real where $a_n=1$. We want to show that The polynomial $g(x)=\sum_{i=0}^{n} ...
16
votes
5answers
2k 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 ...
16
votes
2answers
913 views

$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ ...
16
votes
1answer
479 views

Is every rigid field perfect?

A field is rigid iff its automorphism group is trivial. A field $F$ is perfect iff all irreducibles in $F[x]$ are separable. Is every rigid field perfect?
16
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2answers
1k views

Compactness of $\operatorname{Spec}(A)$

In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact. Now ...
16
votes
2answers
282 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 ...
16
votes
2answers
805 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.
16
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2answers
2k 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 $\mathbb{Z} \subseteq \mathbb{Q}$. ...
16
votes
1answer
545 views

History of Commutative Algebra

There are books on 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. ...
16
votes
1answer
268 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
15
votes
4answers
11k views

A ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
15
votes
2answers
4k views

Why is the localization at a prime ideal a local ring?

I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is ...
15
votes
2answers
801 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?
15
votes
4answers
424 views

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$?

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$? Here tensor product is over the ring $\mathbb Z$ and $\mathbb Z[[X]] $ denotes formal power series over $\mathbb Z$. I think ...
15
votes
1answer
389 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 ...
15
votes
4answers
568 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 ...
15
votes
1answer
926 views

Tensor products of infinite-dimensional spaces and other objects

It has just occurred to me that most of my intuition for tensor products is derived from the special case of finite-dimensional vector spaces, so I'm wondering which properties I've taken for granted ...
15
votes
1answer
253 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 ...
15
votes
3answers
622 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
285 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
195 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
2k views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
14
votes
3answers
3k views

An integral domain whose every prime ideal is principal is a PID

Does anyone has a simple proof of the following fact: An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
14
votes
3answers
2k views

About the localization of a UFD

I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, what about ...
14
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 $A$ ...
14
votes
2answers
671 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 ...
14
votes
2answers
888 views

Hom and tensor with a flat module

Let $A$ be a commutative noetherian ring. Let $M, N$ be $A$-modules, and assume that $M$ is finite over $A$. Let $P$ be a flat $A$-module. Is it true that there is an isomorphism ...
14
votes
2answers
330 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 ...
14
votes
3answers
645 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} ...
14
votes
3answers
689 views

Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff

In the book of Atiyah and MacDonald, I was doing exercise 3.11. One has to show that for a ring $A$, the following are equivalent: $A/\mathfrak{N}$ is absolute flat, where $\mathfrak{N}$ is the ...
14
votes
4answers
2k views

Intuitive explanation of Nakayama's Lemma

Nakayama's lemma states that given a finitely generated $A$-module $M$, and $J(A)$ the Jacobson radical of $A$, with $I\subseteq J(A)$ some ideal, then if $IM=M$, we have $M=0$. I've read the proof, ...
14
votes
2answers
4k views

One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I'm beginning to learn some grothendieck's algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one ...
14
votes
3answers
870 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 ...
14
votes
3answers
562 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
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 ...