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

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2
votes
1answer
74 views

Extension of two-generated ideals in ring extensions

Let $S$ be a subring of a commutative ring $R$ and $a,b,c,d$ are all in $S$. If ideals $(a,b) = (c,d)$ in R, does $(a, b) = (c,d)$ still hold in $S$? If not, what would be a necessary condition on $R$ ...
5
votes
3answers
435 views

Stalk of structure sheaf on fiber product of schemes

Let $X,Y$ be schemes, not necessary separated. Let $f:X\to Y$ be a morphism of schemes and $f(x)=y~,x\in X,~y\in Y$. Then we have the scheme theoretic fiber over $y$, i.e. $f^{-1}(y)=X\times_{Y} ...
0
votes
1answer
100 views

$u\in R$ is a unit iff $u+x$ is a unit for all $x\in \mathcal{N}(R)$

Let $R$ be a commutative ring with identity and denote by $\mathcal N(R)$ its nilradical. It is known that an element $u\in R$ is a unit if and only if $u+x$ is a unit for all $x\in\mathcal N(R)$. In ...
4
votes
2answers
134 views

$\operatorname{Spec}A$ and $T_1$ separation axiom

Let $A$ be a commutative ring. Why $\operatorname{Spec}A$ almost never satisfies the $T1$-separation axiom (Matsumura, Commutative Ring Theory, p.25)?
6
votes
2answers
135 views

Kernel of the product $A\otimes A\rightarrow A$

Let $R$ be a commutative ring with $1$ and let $A$ be a commutative $R$-algebra. We view $A\otimes A$ also as $R$-algebra, the multiplication on generatrs being given by $(a\otimes b)(a'\otimes b')= ...
1
vote
1answer
76 views

Surjectivity in little diagram

Given the following commutative diagram of exact sequences $$ \begin{array} & & 0 & 0 & 0 &\\ & \downarrow & \downarrow & \downarrow &\\ 0 \rightarrow & A ...
5
votes
2answers
290 views

How can we prove this Isomorphism?

How can we prove that $X$ is isomorphic to $Y$? Note: all rows and columns are exact and diagram is commutative. If we can do the following transformation such that ...
3
votes
2answers
255 views

What is a “normalized valuation” corresponding to a valuation ring?

I encountered the phrase "normalized valuation" similar to the following: Let $A_i$ be the valuation ring $k[x_1,...,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by ...
5
votes
0answers
142 views

Is this ring a UFD [duplicate]

Possible Duplicate: Ring of functions that are polynomials in $\cos t$ and $\sin t$, with real coefficients Let $S=\mathbb R[X,Y]/(X^2+Y^2-1)$. Is the ring $S$ a UFD? We obviously have a ...
4
votes
2answers
135 views

$x^2=y^2=(x+y)^2 = 0$ but $xy \neq 0$?

I'm looking for an example of a commutative ring with two elements $x,y$ such that $x^2=y^2=(x+y)^2=0$, and $xy \neq 0$. Obviously, $2$ must be a zero divisor in such a ring. I don't think products of ...
2
votes
3answers
475 views

Extending a homomorphism of a ring to an algebraically closed field

Let $A$ be a subring of a ring $B$. Suppose $B$ is integral over $A$. Let $\Omega$ be an algebraically closed field. Then every homomorphism $\psi\colon A \rightarrow \Omega$ can be extended to a ...
-3
votes
2answers
344 views

Hartshorne Exercise II. 3.19 (b)

How do we prove the following exercise of Hartshorne? Let $A$ be a subring of an integral domain $B$. Suppose $B$ is a finitely generated $A$-algebra. Let $b$ be a non-zero element of $B$. Then there ...
1
vote
2answers
103 views

Commutative Rings and Ideals

I'm having trouble understanding some of the subset theory for commutative rings. Is it possible to sandwich a non-ideal within 2 ideals of commutative rings? So for example, is it possible to ...
1
vote
2answers
72 views

Commutative Rings Inside Commutative Rings with Field properties

As I was working through an algebra textbook, I noticed that a field $A$ is a commutative ring. But is it possible for $A \subset B \subset C$ where $A$, $C$ are fields and $B$ is not (and all of them ...
3
votes
1answer
497 views

Explicit example of Koszul complex

Let $R$ be a Nothearian commutative ring and $x$ and $y$ two elements in $R$. I want to construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes $$C_2=0\to ...
2
votes
2answers
413 views

Maximal ideals in multivariate polynomial rings

Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to ...
1
vote
1answer
166 views

Linear projective varieties

Let $Y\subset\mathbb{P}_{k}^{n}$ be a projective variety. We say that Y is a linear subvariety if $I(Y)$ can be generated by linear polynomials. Now how I should show that $Y\subset\mathbb{P}_{k}^{n}$ ...
3
votes
4answers
513 views

Noetherian module implies Noetherian ring?

I know that a finitely generated $R$-module $M$ over a Noetherian ring $R$ is Noetherian. I wonder about the converse? I believe it to be false and I am looking for counterexamples. Also I wonder if ...
4
votes
2answers
99 views

Show there exists a finite flat morphism

If $f \in k[x_1,...,x_n]$ is irreducible then show there is a finite flat morphism $k[x_1,...,x_{n-1}] \to k[x_1,...,x_n]/(f)$ (i.e. $k[x_1,...,x_{n}]/(f)$ is finitely generated and flat as a module ...
4
votes
1answer
159 views

About prime ideals partial derivatives of polynomials

Given a polynomial $f(x_1,... ,x_n)\in \mathbb{C}[x_1, ... ,x_n]$, we can formulate its (formal) partial derivative with respect to each of the $x_i$, say $f_{i}$. If $f\in \mathfrak{p}$ and $f_{i}\in ...
3
votes
1answer
87 views

Calculation of dimension of Socle

Let $S=k[[t^3,t^5,t^7]]$ be a formal power series over field $k$.I wanna know why $$\dim_k \operatorname{Soc}(S/t^3S)=2?$$.($\dim_k$ means dimension as $k$-vector space.) background: ...
2
votes
1answer
97 views

Ring containing a Dedekind ring

Suppose I have two domains, $A\subset B$, where $A$ is Dedekind and $\operatorname{Frac}(A)=\operatorname{Frac}(B)$. I also know that $B$ is both integrally closed and has height $1$. Is $B$ ...
7
votes
1answer
99 views

Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
12
votes
5answers
832 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 ...
5
votes
2answers
301 views

Which field is this quotient of a local ring by its maximal ideal?

Let $p\in\mathbb{Z}$ be a prime number, $\mathfrak{p}\subset \mathbb{Z}$ be the prime ideal it generates and let $\mathbb{Z}_{\mathfrak{p}}$ be the localization of $\mathbb{Z}$ at $\mathfrak{p}$, i.e. ...
3
votes
1answer
43 views

Null map between exact sequences

Is it true that if I have exact sequences of abelian groups $0 \rightarrow A\rightarrow B \rightarrow C\rightarrow 0$ and $ A_1\rightarrow B_1 \rightarrow C_1 $ (exact only in the middle) and ...
2
votes
3answers
215 views

Atiyah - Macdonald Exericse 9.7 via Localization

I am trying to show that the quotient of a Dedekind domain $A$ by an ideal $\mathfrak{a}$ is a PIR (principal ideal ring). Now by using the Chinese Remainder Theorem and the fact that a direct product ...
5
votes
1answer
480 views

Constructing Idempotent Generator of Idempotent Ideal

Exercise 2.1 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a commutative ring and $I$ an ideal that is finitely generated and $I=I^2$. Then $I$ is generated by an idempotent." ...
2
votes
1answer
160 views

Non-Free Finitely Generated Injective Modules over a Local Ring

I was wondering if someone could be so kind as to provide an example of a local ring $ (R,\frak{m}) $ and a non-free finitely generated injective module over $ R $. Thank you very much! I tried ...
3
votes
1answer
186 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
10
votes
0answers
202 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
6
votes
1answer
208 views

Is there any deep connection between algebraic topology and homological algebra on rings?

There is a deep connection between algebraic topology and homological algebra on groups. A group $G$ can be interpreted as the fundamental group of a covering space $Y \rightarrow X$. (Co)Homology ...
3
votes
1answer
663 views

UFD implies noetherian?

It is easy to show that a PID must be noetherian. My question is: Does UFD imply noetherian? If not, is there an easy counterexample? I apologize if this turns out to be a simple question. ...
2
votes
0answers
92 views

Kähler differentials, smooth algebras (one specific question)

Let $A$ be a commutative algebra (of essentially finite type, perhaps I have to add other reasonable assumptions) over a field, and I assume that $\Omega^1(A)$ is a projective module. Then the ...
4
votes
2answers
112 views

Is the category of quasi-coherent $\mathcal{O}_X$-algebras cocomplete?

Let $X$ be a scheme. Is the category of quasi-coherent (commutative) $\mathcal{O}_X$-algebras cocomplete?
5
votes
2answers
108 views

How to show $\bigcap_{m \textrm{: maximal ideal}} A_m=A$?

$A$ is an integral domain. For every maximal ideal $m$ in $A$, consider $A_m$ as a subring of the quotient field $K$ of $A$. Show $\bigcap A_m=A$, where the intersection is taken over all maximal ...
2
votes
0answers
104 views

Geometric interpretations of several algebraic concepts

I would like to have some geometric intuition for - Noetherian rings/modules - Local rings - Projective modules - Injective modules As an illustration of what I am looking for, I was told once that ...
3
votes
1answer
335 views

Using Nakayama lemma to prove that surjective implies injective.

$A$ is a commutative ring, and $f:M\rightarrow M$ is an endomorphism of $A$-modules which is surjective. If I know that $M$ is finitely generated, I want to prove that $f$ is also injective. ...
2
votes
0answers
68 views

Reduction of maximal graded ideal

Let $R$ be a commutative graded ring, $m$ be its graded maximal ideal, $M$ be a finitely generated graded module over $R$. A homogeneous ideal $I\subseteq m$ is a $M$-reduction of $m$ if ...
3
votes
1answer
336 views

Automorphisms of an affine line over finite fields

Let $k$ be an infinite field, and consider the affine line $\mathbb{A}_k^1$ over $k$. We know that every isomorphism $\varphi:\mathbb{A}_k^1\longrightarrow\mathbb{A}_k^1$ is of the form ...
6
votes
4answers
287 views

Is this a property of an integral domain that is not a field?

I am working on a specific problem and I've almost got it solved. To solve it, however, I need to prove one last claim (if it is even true): Consider an integral domain $R$ that is not a field. ...
2
votes
0answers
132 views

Divisorial ideal of a Krull domain

Now I try to do exercise 12.4 in the book "Commutative ring theory" by H. Matsumura. Let $A$ be a Krull domain, $I\subseteq \mathfrak p$ and $\mathfrak p$ is a height $1$ prime ideal of $A$. I don't ...
0
votes
1answer
104 views

Question about extensions of homomorphisms

I have difficulty understanding the proof of Theorem 3.2 in Lang's Algebra Chapter VII. Let $A$ be a subring of a field $K$ and let $x\in K, x\neq 0$. Let $\phi:A \rightarrow L$ be a ...
1
vote
1answer
82 views

Elementary questions about regular rings and Zariski tangent spaces

So I've got 3 rather related questions, which all seem to be true, except maybe the third. I'm asking because I remember thinking about this in the past and encountering a difficulty with all 3. ...
2
votes
0answers
71 views

ring of invariants for E6 singularity

As an exercise for myself, I've been trying for some time to calculate the coordinate ring of the $E6$ surface singularity. That is, I have a 2d representation of the binary tetrahedral group ...
0
votes
2answers
502 views

Localization at Maximal Ideals

Suppose $A$ is a commutative ring with $1\neq0$ satisfying the property that $A_\mathbf{m}$ has no nonzero nilpotent elements for any maximal ideal $\mathbf{m}$, where $$A_\mathbf{m}=S^{-1}A\quad ...
4
votes
1answer
242 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} ...
6
votes
1answer
211 views

Questions about subalgebras of finitely generated $k$-algebras

Let $k$ be a field (if necessary assume $k$ to be algebraically closed). Let $A$ be a finitely generated $k$-algebra and let $B$ be a subalgebra of $A$. Remark that $B$ doesn't have to be noetherian, ...
3
votes
1answer
139 views

Exercise from Matsumura about DVRs

Another result I would really appreciate some help with: Suppose $R$ is a DVR and let $K$ be its field of fractions. Let $L$ be a finite extension of $L$. Prove that any valuation domain inside of ...
4
votes
1answer
292 views

Algebraic independence and dimension of a variety

A set of polynomials $\{f_1,\ldots,f_m\}$ in $k[x_1,\ldots,x_n]$ are algebraically independent over $k$ iff for all polynomials $p \in k[y_1,\ldots,y_m]$, $p(f_1,\ldots,f_m) = 0$ implies that $p = 0$. ...