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

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15
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676 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
5answers
2k views

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
15
votes
1answer
179 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
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3answers
448 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
272 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
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1answer
138 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
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2answers
998 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 ...
14
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1answer
492 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 ...
14
votes
3answers
443 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
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1answer
222 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
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2answers
159 views

If $M\otimes N=R^n$ need $M$ be projective?

So if over a commutive ring, $R$, we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ be finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
14
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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
429 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
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2answers
167 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
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5answers
2k 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 ...
13
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2answers
487 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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5answers
1k 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
393 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
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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 ...
13
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2answers
301 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
480 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
13
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4answers
513 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
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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 ...
13
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3answers
597 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
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5answers
559 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
293 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
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1answer
371 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
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1answer
499 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
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1answer
135 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
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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 ...
12
votes
3answers
500 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
2answers
334 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
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2answers
354 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
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2answers
515 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
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1answer
328 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. ...
12
votes
5answers
763 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
441 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
181 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
143 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
154 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
187 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
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1answer
366 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 ...
11
votes
2answers
1k views

Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$

I want to understand the tensor product $\mathbb C$-algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$. Of course it must be isomorphic to $\mathbb{C}\times\mathbb{C}.$ How can one construct an ...
11
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2answers
417 views

Examples demonstrating that the finitely generated hypothesis in Nakayama's lemma is necessary

Recall that Nakayama's lemma states that Let $R$ be a commutative ring with unity, and let $J$ be the Jacobson radical of $R$ (the intersection of all the maximal ideals of $R$). For any finitely ...
11
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3answers
450 views

Simple example of non-arithmetic ring

Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?
11
votes
2answers
698 views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
11
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2answers
781 views

A non-noetherian ring with all localizations noetherian

If for a ring $A$ every localization $A_\mathfrak{p}$ by a prime $\mathfrak{p}\subseteq A$ is noetherian, is it true that $A$ is noetherian? I believe not but I can't find a good counterexample.
11
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2answers
262 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
11
votes
3answers
1k 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 ...
11
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2answers
229 views

Does inclusion of a ring into a polynomial ring induce a closed map on prime spectra?

Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by ...