Questions about commutative rings, their ideals, and their modules.
52
votes
4answers
2k 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 $S$ ...
38
votes
5answers
1k 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 ...
31
votes
4answers
1k views
Golden Number Theory
The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers ...
28
votes
3answers
2k 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 ...
23
votes
0answers
362 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 ...
22
votes
1answer
497 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 ...
21
votes
5answers
1k 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 ...
21
votes
4answers
377 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 ...
21
votes
1answer
442 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 ...
21
votes
0answers
329 views
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 nontrivial ...
20
votes
1answer
366 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.
19
votes
3answers
936 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 ...
19
votes
1answer
346 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?
18
votes
3answers
306 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,
...
18
votes
2answers
409 views
What are exact sequences, metaphysically speaking?
Why is it natural or useful to organize objects (of some appropiate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
18
votes
2answers
362 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 ...
18
votes
1answer
329 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 ...
18
votes
0answers
258 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 ...
17
votes
5answers
2k 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 ...
17
votes
4answers
753 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' ...
17
votes
2answers
533 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 ...
17
votes
3answers
247 views
Bound on nilpotency index of endomorphisms
Let $A$ be a Noetherian algebra (commutative with 1 over $\mathbb{Z}$) and $M$ a finitely-generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism ...
17
votes
2answers
341 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 ...
16
votes
9answers
2k views
Motivation for Tensor Product
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 ...
16
votes
2answers
379 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, ...
16
votes
3answers
361 views
A non-Noetherian Lasker ring
A Lasker ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Lasker ring (as an easy consequence of the ...
16
votes
4answers
1k 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: ...
16
votes
1answer
360 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} ...
15
votes
4answers
662 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 ...
15
votes
2answers
508 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 ...
15
votes
2answers
894 views
A tensor product of a 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 ...
15
votes
2answers
427 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 ...
15
votes
1answer
374 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
1answer
257 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, ...
14
votes
3answers
234 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 ...
14
votes
3answers
315 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
1answer
360 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 ...
14
votes
3answers
319 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 ...
14
votes
0answers
104 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 ...
13
votes
1answer
364 views
What is Spec $\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$?
Let $X = \textrm{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 possible to ...
13
votes
2answers
306 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
votes
4answers
528 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 ...
13
votes
2answers
865 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, ...
13
votes
1answer
251 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
415 views
Exercise 2.17(d) of Eisenbud's Commutative Algebra text
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
2answers
146 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 ...
12
votes
3answers
367 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 ...
12
votes
3answers
664 views
Motivation for Eisenstein Criterion
I have been thinking about this for quite sometime.
Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a commutative unique factorization domain $R$, say
$$f(x)=a_0 + ...
12
votes
3answers
430 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 ...
12
votes
5answers
271 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 $A$ ...
