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

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303
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A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
179
votes
0answers
6k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
88
votes
4answers
3k views

Does $R[x] \cong S[x]$ imply $R \cong S$?

This is a very simple question but I believe it's nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and ...
57
votes
3answers
6k views

What is the Tor functor?

I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor. I have tried Googling the term, but I ...
50
votes
5answers
2k views

How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
43
votes
8answers
4k views

Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean ...
41
votes
1answer
4k views

Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$ $(f(X))$, where ...
39
votes
5answers
3k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
37
votes
2answers
1k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
36
votes
0answers
950 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
35
votes
1answer
535 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
35
votes
1answer
615 views

Is Frobenius the only magical automorphism?

The Frobenius automorphism is special because the $p$-power map makes sense in any characteristic $p$ ring, which allows us to canonically extend the Galois-theoretic Frobenius to any such ring. I ...
33
votes
9answers
5k views

Why is the tensor product important when we already have direct and semidirect products?

Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do ...
32
votes
4answers
1k views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
32
votes
3answers
4k views

Reference request: introduction to commutative algebra

My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura ...
32
votes
3answers
3k views

Why does a minimal prime ideal consist of zerodivisors?

Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. This can be proved using localization, when $A$ is noetherian: ...
30
votes
3answers
1k views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
29
votes
6answers
2k views

Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. ...
28
votes
0answers
479 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
27
votes
4answers
3k views

Commutative non Noetherian rings in which all maximal ideals are finitely generated

In commutative rings we have the following Theorem. $R$ is Noetherian if and only if each prime ideal of $R$ is finitely generated. From this Theorem I am looking for commutative rings $R$ in which ...
25
votes
4answers
1k views

Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
25
votes
4answers
2k views

Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' ...
25
votes
3answers
654 views

A Laskerian non-Noetherian ring

A Laskerian ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Laskerian ring (as an easy consequence of the ...
25
votes
1answer
719 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 ...
24
votes
2answers
3k views

Tensor products commute with direct limits

This is Exercise 2.20 in Atiyah-Macdonald. How can we prove that $\varinjlim (M_i \bigotimes N) \cong (\varinjlim M_i) \bigotimes N$ ? Atiyah gives a suggestion, he says that one should obtain a map ...
23
votes
2answers
816 views

The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
23
votes
1answer
720 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
22
votes
3answers
477 views

My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes: Let $R$ be a ...
22
votes
3answers
468 views

$\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$?

In the course of working on an exercise in Atiyah-MacDonald (exercise 3 on p. 31), I've come to the belief that, for $A$ an arbitrary commutative ring and $M,N$ arbitrary $A$-modules, ...
22
votes
3answers
2k views

If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ ...
22
votes
3answers
905 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
22
votes
1answer
465 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 ...
22
votes
1answer
642 views

Why is the Hessian of an irreducible polynomial not zero?

Let $k$ be an algebraically closed field, $\operatorname{char}k=0$, $F$ be an irreducible homogeneous polynomial of degree$>1$ in $k[X,Y,Z]$, and ...
22
votes
1answer
479 views

Hilbert's original proof of basis theorem

Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for ...
21
votes
2answers
2k views

A tensor product of power series

Let $k$ be a field. I am wondering if there is an easy description of the ring $$k[[x]] \otimes_{k[x]} k[[x]]$$ that is the tensor product of the power series ring $k[[x]]$ with itself over the ring ...
21
votes
1answer
603 views

The tensor product of two Artinian modules is Artinian

user$xxxxx$ posted (and then deleted) the following question which I think deserves to be here: Prove that the tensor product of two Artinian modules is Artinian.
21
votes
1answer
506 views

functoriality of derivations

I seem to have problems understanding algebraically why given a map of manifolds $f: M \to N$ we get a bundle map $TM \to f^*TN$. Now, fiberwise it's all good. But I do not understand how to define ...
20
votes
4answers
7k views

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

Free modules are projective, and projective modules are direct summands of free modules. Are there examples of projective modules that are not free? (I know this is not possible for modules of ...
20
votes
4answers
1k views

Why are ideals more important than subrings?

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 ...
20
votes
1answer
603 views

What is $\operatorname{Spec}\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$?

Let $X = \operatorname{Spec} \mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$. I would like to describe $X$ set-theoretically. My questions are: Can one explicitly say what the elements in $X$ are? Is it ...
20
votes
2answers
743 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
20
votes
2answers
1k views

Atiyah-Macdonald Exercises 5.16-5.19

I have solutions to Exercises 5.16–5.19 in Atiyah–Macdonald's Introduction to Commutative Algebra, but not in the order desired; I find myself needing later exercises to do earlier ones, ...
20
votes
1answer
633 views

Are finitely generated projective modules free over the total ring of fractions?

Let $Q(A)$ be the total ring of fractions of a commutative reduced non-noetherian ring $A$. Let $P$ be a finitely generated projective module over $Q(A)$ which is of constant rank (i.e. locally free ...
19
votes
5answers
6k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
19
votes
3answers
700 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 ...
19
votes
5answers
3k views

Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1. In a ring $A$, let $\Sigma$ be the set of all ideals in which every ...
19
votes
4answers
3k 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 ...
19
votes
1answer
3k 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 ...
19
votes
1answer
824 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 ...
19
votes
2answers
648 views

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...