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

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3
votes
3answers
419 views

Hilbert-Poincaré Series of Finite-Dimensional Graded Algebras

Suppose I have two finite-dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbb{C}$-algebras $A = \bigoplus_{k \geq 0} A_{k}$ and $B = \bigoplus_{k \geq 0} B_{k}$ with Hilbert-Poincaré series, $P_{A}(t) = ...
1
vote
1answer
114 views

Finite morphism and Proj

Let $A \subset B$ be a finite extension of graded rings (so that $B$ is finite as a graded $A$-module). There is an induced finite morphism $\operatorname{Proj} B \to \operatorname{Proj} A$. Is this ...
3
votes
4answers
579 views

Radical ideal of $(x,y^2)$

How does one show that the radical of $(x,y^2)$ is $(x,y)$ over $\mathbb{Q}[x,y]$? I have no idea how to do, please help me.
16
votes
1answer
462 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 ...
11
votes
3answers
652 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
7
votes
2answers
322 views

Noetherian integral domain such that $m/m^2$ is a one-dimensional vector space over $A/m$

I am having trouble doing the following question (I'm studying for quals, it isn't homework) If $A$ is a noetherian integral domain such that for every maximal $m\subset A$, the quotient $m/m^2$ is a ...
2
votes
1answer
134 views

Ideal structure of a ring, flat over a PID

I know that the following question is somewhat vague; I'm trying to understand what flatness "buys". Suppose $A, B$ are rings (commutative with identity) with $A$ a PID. Suppose we are given a flat ...
11
votes
3answers
473 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)?
7
votes
2answers
845 views

Direct summand of a free module

Let $M$, $L$, $N$ be $A$-modules and $M=N\oplus L$. If $M$ and $N$ are free, is $L$ necessarily free?
5
votes
2answers
193 views

An integral ring extension that $B/M$ is not separable over $A/m$

I am trying to find an example like this. Let $A$ be integrally closed in its field of fraction $K$ and $L$ a finite Galois extension. Let $B$ be the integral closure of $A$ in $L$. Is there a ...
1
vote
1answer
298 views

Complement of saturated set

Let A be an integral domain and $S \subset A$ — its saturated subset (multiplicative subset s.t. if $ab \in S$ then $a \in S$ and $b \in S$). Would you please supply a hint, how to prove that $A ...
1
vote
6answers
765 views

Proof that $\mathbb{Z}$ has no zero divisors

Everyone knows the rules of zero divisors like $$\forall \alpha,\beta\in\mathbb{R}\;:\;\alpha\cdot\beta = 0\Rightarrow\alpha=0\vee \beta=0.$$ But how can I prove it for $\mathbb{Z}$? My first try was ...
3
votes
0answers
121 views

Local Coordinate Systems under Integral Extension

Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be an integral extension of regular local rings of dimension $d$ (of course, $\varphi$ is a local homomorphism). Furthermore, assume that $A$ contains ...
0
votes
1answer
175 views

surjection and injection of modules over local ring

If I have a finitely generated module $M$ over a local noetherian ring $(A,m)$, where $m$ denotes the maximal ideal, and $\operatorname{supp}{(M)}=m$, then there exists a surjection $$M\rightarrow ...
10
votes
1answer
616 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 ...
18
votes
2answers
1k 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 ...
11
votes
3answers
411 views

A finite commutative ring with the property that every element can be written as product of two elements is unital

I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a ...
3
votes
1answer
169 views

$M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$

Let $S.$ be a graded ring, finitely generated by degree 1 elements as a $S_0$-algebra. Let $M.$ be a finitely generated graded $S.$-module. There exists a natrual map ...
1
vote
1answer
115 views

When is an extension of a prime also a prime?

Suppose $R$ and $S$ are domains, $S$ is integral over $R$, $R$ is integrally closed, $p_1$ and $p_2$ are primes in the domain $R$, $p_1$ contains $p_2$, $q_1$ is a prime in the domain $S$ and lying ...
5
votes
2answers
182 views

Having trouble with just one line in a proof on why nonzero prime ideals are maximal in a Dedekind domain

http://planetmath.org/?op=getobj&from=objects&name=ProofThatADomainIsDedekindIfItsIdealsAreInvertible In the PlanetMath article above, in the second paragraph of the proof of the first lemma, ...
5
votes
2answers
540 views

A condition for a subgroup of a finitely generated free abelian group to have finite index

Let A be a free Abelian group of finite rank and B be a subgroup of A such that $A=B+pA$ for some prime number p, then how to prove $B$ is a subgroup of finite index in A? And if $A=B+pA$ holds for ...
17
votes
2answers
739 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 ...
3
votes
1answer
311 views

Flat algebras and tensor product

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules. Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$? Thanks
4
votes
1answer
131 views

Is the support of an Artinian module finite?

If $R$ is an Artinian ring then it has finite maximal ideals. If $M$ is an $R$-module Artinian. ($R$ be a commutative Noetherian ring). Then, is $Supp(M)$ finite? Thanks.
3
votes
1answer
146 views

Does the minimal injective resolution have the smallest length?

Let $A$ be a Noetherian (not necessarily local) ring and $M$ a finitely generated $A$-moduel. Is the length of the minimal injective resolution of $M$ always equal to the injective dimension of $M$? ...
1
vote
2answers
267 views

isomorphism between specific generated field and specific quotient ring — gap in a proof

$K'$ is a field extension of $F$, $h\in F[x]$, $h$ is minimal for $u'\in K'$, $F(u')$ is a field generated by $F\cup \{u'\}$, $K'=F(u')$. In [1. XIII. Galois theory. 2. ...
11
votes
2answers
2k 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 ...
22
votes
1answer
581 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 ...
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 ...
5
votes
2answers
462 views

Are the determinantal ideals prime?

I want to prove the determinantal ideals over a field are prime ideals. To be concrete: For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be ...
6
votes
3answers
232 views

Proof that a certain derivation is well defined

I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf Suppose, $R$ is ...
3
votes
1answer
131 views

Every set of $n$ generators of $A^{n}$ is actually a basis

Let $A$ be a commutative ring with $1$. It is a standard result that every set of $n$ generators of the free $A$-module $A^{n}$ is actually a basis. The proof uses tensor products. I was reading a ...
2
votes
1answer
120 views

Regularity ascends from a Noetherian ring to a polynomial or power series ring over it

I am looking for a proof of the following statement: A Noetherian ring $R$ is regular if and only if $R[x]$ is regular if and only if $R[[x]]$ is regular. I am trying to understand the properties ...
4
votes
1answer
225 views

Integral extensions: one prime lying over implies equal localization

Here's a problem from Matsumura's book "Commutative ring theory" page $69$. Let $A$ be a ring and let $A \subset B$ be an integral extension, and $\mathfrak{p}$ a prime ideal of $A$. Suppose that $B$ ...
5
votes
1answer
155 views

Grothendieck spectral sequence

given functors $F,G$, left exact, with as good properties as you want we have a spectral sequence $R^p F\circ R^q G$ abutting to $R^{p+q}(F\circ G)$. I am looking for an analogous for a "mixed ...
0
votes
1answer
67 views

Dimensions of modules of the maximal compact subrings of locally compact fields

I have checked the list of similar titles, proposed by the site. I hope this is not a repetition. This question arises from a proof of a proposition in the book Basic Number Theory, as follows. ...
1
vote
1answer
125 views

Algebra over a ring

Could someone point me to a proof which shows that an algebra over a ring can be presented as a quotient of a polynomial ring (in possibly infinitely many variables).
4
votes
3answers
251 views

Localization of prime ideals

Let $A$ be a commutative ring with $1$. Suppose that $P \subseteq Q$ are prime ideals in $A$ and that $M$ is an $A$-module. Prove that the localization of the $A$-module $M_{Q}$ at $P$ is the ...
0
votes
2answers
80 views

Injective modules and ring homomorphisms

If there is a ring homomorphism $A\rightarrow B$ and if $Q$ is an injective $A$-module, is it true that $Q\otimes_A B$ is an injective $B$-module? I don't think it's true but can't think of a ...
2
votes
1answer
238 views

Smooth ring maps and the module of differentials

Suppose $A$ and $B$ are commutative Noetherian rings, and $A \to B$ is a finite type smooth map. Then it is well know that the module of Kähler differentials, $\Omega^1_{B/A}$ is a projective module ...
4
votes
1answer
112 views

What are the sections of $\mathbb{C}\rightarrow \mathbb{C}[[x,y]]^{alg}[\frac{1}{xy}]$?

I've been thinking about weird rings recently, and I couldn't answer the following question to myself: What are the sections of the inclusion $\mathbb{C}\rightarrow ...
6
votes
2answers
654 views

Computing ideal intersections in polynomial rings

Suppose, $R=k[x_1,...,x_n]$ and $I,J,A,B,C,D$ are ideals in $R$. Suppose, I can write $A,B,C,D$ explicitly in terms of generators and I can also compute $A\cap B$ explicitly in terms of generators. It ...
2
votes
1answer
338 views

Tensor products and polynomials with coefficients in a module

This is exercise $6$ from Atiyah's and Macdonald's book. Let $M$ be an $A$-module and let $M[x]$ be the set of all polynomials in $x$ with coefficients in $M$. Then $M[x]$ has structure of ...
4
votes
1answer
429 views

What are the relations between the Koszul complex and the minimal free resolution?

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $F.$ the Koszul complex of a minimal system of generators of $\mathfrak{m}$. Let $G.$ be the minimal free resolution of $k$. In which cases they ...
0
votes
2answers
147 views

Ring automorphism in an algebraically closed field

Let $K$ be an algebraically closed field and let $\mathfrak{m}$ be a maximal ideal of $K[x_{1},..,x_{n}]$. How to show there is a ring automorphism $f$ of $K[x_{1},..,x_{n}]$ such that: ...
4
votes
3answers
258 views

Support and tensor product doubts

Some questions: 1) This is proposition $3.5$ , page $39$ of Atiyah's and Macdonald's book. Let $M$ be an $A$-module. Then $S^{-1}A \otimes_{A} M \cong S^{-1}M$ as $S^{-1}A$-modules. So the idea is ...
4
votes
1answer
182 views

Must the projective closure of a closed subset of affine space have points at infinity?

let $k$ be an algebraic closed field. For $I\subseteq k[x_1,\ldots,x_n]$, we donote $I^*$ to be the ideal generated by the set $\{F^*|F\in I\}$, here $F^*=x_{n+1}^{deg ...
1
vote
1answer
222 views

Fibre product is Noetherian

Let $A$ and $B$ be Noetherian rings and $f: A \rightarrow C$ and $g: B \rightarrow C$ ring homomorphisms. If both $f$ and $g$ are surjective show $\{(a,b) \in A \times B: f(a)=g(b)\}$ is a Noetherian ...
4
votes
4answers
299 views

Every set of $n$ generators is a basis for $A^{n}$

This is problem $15$ from Chapter $3$ of Atiyah's and Macdonald's book. Let $A$ be a ring let $F$ be the $A$-module $A^{n}$. Show that every set of $n$ generators is a basis of $F$. Here's the hint: ...
3
votes
1answer
144 views

Finding basis of a free $\oplus_{i=1}^m k$ module in a subspace

I come across this question as I consider a problem dealing with semilocal rings. Suppose that $k$ is a field, $R=\oplus_{i=1}^m k$ is a finite $k$-algebra via the diagonal embedding $k\to R$. Let ...