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

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3
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1answer
66 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no $Q$ prime ideal such that $0 \subsetneq Q \subsetneq P$. ...
0
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0answers
13 views

How to construct a unique valuation for $k\left(T_{i}\right)_{i\in\mathbb{N}}$ in $\mathbb{Z}^{\left(\mathbb{N}\right)}$?

Let $k$ be a field and $\left(T_{n}\right)_{n\in\mathbb{N}}$ indeterminates over $k$. Let $K=k\left(T_{n}\right)_{n\in\mathbb{N}}$ and $\varGamma:=\mathbb{Z}^{\left(\mathbb{N}\right)}$ the abelian ...
0
votes
1answer
39 views

Artin local ring [duplicate]

I have studied "structure theorem for Artin rings" which states "An Artin ring $A$ is unique a finite direct product of Artin local rings". Let $A$ be Artin ring. By Chinese remainder theorem, $A ...
0
votes
1answer
53 views

Is $A \rightarrow S^{-1} A$ epi?

this question must be the most stupid I have ever asked. If $A$ is a commutative ring and $S$ a multiplicative subset the usual inclusion induces a homeomorphism onto the image $\text{Spec} ...
0
votes
0answers
30 views

Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
3
votes
1answer
60 views

Two elements in a non-integral domain which are not associates but generate the same ideal

Let $\mathbb{K}$ be a field. Let $R$ be the quotient ring $\mathbb{K}[x,y]/(xy^{2})$. Let $\bar{x}$ be the class of $x$ in $R$ (i.o.w. $\bar{x}=x+(xy^{2}))$. Prove that $\bar{x}$ and ...
4
votes
1answer
28 views

Irreducible radical ideals are prime

Assume $R$ is a commutative ring and $I$ is a nonzero proper ideal of $R$ satisfying: $(1)$ If $I_1$ and $I_2$ are ideals such that $I = I_1 \cap I_2$, then $I = I_1$ or $I = I_2$; $(2)$ If $a^n ...
0
votes
0answers
31 views

Length of chain of prime ideals in polynomial ring

Let $B=A[x_1,...,x_n]$ be a ring of polynomials over the ring $A$, $P$ be a prime ideal in $A$. Suppose that we have the chain $Q_0\subset Q_1\subset ... \subset Q_k$ of strictly embedded prime ideals ...
1
vote
1answer
29 views

The non flat module $\mathbb{Z}/m$.

A general result states that an $R$-module $M$ is flat if and only if $I\otimes_R M \simeq IM$ for all ideals $I\subset R$. However, there is something I don't understand. Let $R = \mathbb{Z}$, and ...
8
votes
2answers
231 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
1
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2answers
49 views

examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
0
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0answers
29 views

What is the kernel of $I/I^2 \to \Omega_{\mathbb P^{n}/k} \otimes \mathcal O_X$?

Recall that if $X \subset \mathbb P^n$ is a smooth projective variety, we have the conormal sequence of locally free sheaves on $X$ (here $I$ is the ideal sheaf of $X$): $$ I/I^2 \xrightarrow{\delta} ...
0
votes
1answer
48 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
2
votes
2answers
57 views

Two ways to localize a ring using a prime ideal

I was reading the part about localization of the Introduction to Commutative Algebra of Atiyah-MacDonald and I have a question I was not able to solve. Let $R$ be a commutative ring with unit $1$ and ...
2
votes
1answer
47 views

An easy question about fractional ideals…

Let $A$ be an integral domain and $K$ its field of fractions. If $M$ is a non-zero fractional ideal of $A$, then $$N=\{x \in K : xM \subseteq A\}$$ is also a fractional ideal of $A$. The proof I am ...
1
vote
1answer
29 views

Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$

Suppose $R$ is integral domain and $K$ is the fraction field of $R$. Please give me an example of $a \in K-R$ but there exists $n\in\mathbb{N}$, $a^n\in R$.
2
votes
1answer
41 views

Proving $C\otimes_A\Omega^1_{A/R} \cong \Omega^1_{C/B}$

I am completely stuck on this so any help would be great. Let $R$ be a commutative ring and let $A$ and $B$ be $R$-algebras. Let $C:=A\otimes_RB$. Show that $C\otimes_A\Omega^1_{A/R} \cong ...
2
votes
2answers
44 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
3
votes
1answer
54 views

Motivation and examples for ramification

I started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...
1
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0answers
34 views

Localization at a prime and direct limits

Let $R$ be a commutative ring with $1 \neq 0$ and let $P \subset R$ be a prime ideal. Apparently we have $$\varinjlim\limits_{f \in R \setminus P} R_f \cong R_P$$ where $R_f$ the the localization of ...
1
vote
1answer
15 views

How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$?

I'm trying to solve exercise 4.7 in Reid's UCA: "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$." I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus ...
4
votes
1answer
64 views

If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: $I_n$ is generated by a (finite) number of polynomials which ...
0
votes
0answers
36 views

a math-software that can compute analytic spread

I want to compute "analytic spread" . So I need a math-software that can compute it. can anyone help please? Here is the definition:
1
vote
1answer
52 views

Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle?

Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} ...
1
vote
2answers
55 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
2
votes
1answer
38 views

Exercise of commutative algebra, rational functions.

This exercise is of my weekly newsletter of the subject of commutative algebra. My knowledge is restricted to the book of William Fulton, Algebraic Curves. I need help to solve it, any hints. ...
2
votes
1answer
53 views

Counterexample for the infinitely many primes between two primes in a Noetherian ring

Consider the following Proposition: Proposition: Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many ...
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votes
2answers
69 views

Ring with nested prime ideals [closed]

If $n>1$ is there a (commutative with identity) ring with Krull dimension $n$ and only $n+1$ prime ideals?
0
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1answer
30 views

A condition that an algebraic set is irreducible.

From the book by Kenji Ueno, Algebraic Geometry 1. From Algebraic Varieties to Schemes: "If an algebraic set $V(J)$ is reducible, it can be expressed as: $$(1.8)\quad V(J)= V(J_1)\cup V(J_2), \ ...
0
votes
0answers
37 views

Computing homomorphisms between extensions of modules

Suppose we have two exact sequences of $R$-modules ($R$ is a commutative ring) $$0\rightarrow M_0\rightarrow F\rightarrow M_1\rightarrow0$$ $$0\rightarrow N_0\rightarrow G\rightarrow ...
0
votes
0answers
33 views

For a ring homomorphism, why does $f$ induces a homeomorphism from $SpecB$ onto the closed subset $V(\ker f)$ of $SpecA$.

Let $\varphi : A \rightarrow B$ be a ring homomorphism. Then we have a map of sets $Spec(\varphi):Spec(B) \rightarrow Spec(A)$ defined by $p \mapsto \varphi^{-1}(p)$ for every $p \in SpecB$. ...
1
vote
1answer
58 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
0
votes
0answers
29 views

How to calculate the multiplicity of semigroup ring of dimension one?

Let $k$ be a field and $R=k[t^{a_1},...,t^{a_n}]$ such that $0<a_1<a_2<\cdots<a_n$ are integers. Is $a_1$ the multiplicity of $R$? Why?
1
vote
1answer
23 views

$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
1
vote
1answer
51 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
1
vote
1answer
26 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
0
votes
0answers
24 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
-1
votes
1answer
38 views

Question about notation on ideals

If $R$ is a commutative ring and $a,b \in R$ then $(a)+(b)=(a,b)= \{xa+yb : x,y \in R \}$, however if $I$ is an ideal of $R$ then what is $(I,a)$? My guess is $(I,a)=\{hg + xa : h,x \in R, g \in I ...
1
vote
1answer
24 views

Comparing an ideal and its saturation

Let $S = k[x_0,x_1,\ldots,x_n]$ with its usual grading and let $I \subset S$ be a homogeneous ideal not containing $S_+ = (x_0,x_1,\ldots,x_n)$. We define the saturation of $I$ to be the homogeneous ...
1
vote
0answers
30 views

proof of Proposition 3.3.18 in Bruns and Herzog

This set of questions pertains to the proof of Proposition 3.3.18(b) in Bruns and Herzog, Cohen-Macaulay Rings: Question 1: It seems to me that under the hypothesis (a) of the theorem, the ...
0
votes
0answers
42 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
2
votes
1answer
51 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
1
vote
1answer
34 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...
0
votes
1answer
55 views

Example of Gorenstein local ring of dimension 1

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
1
vote
1answer
51 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
1
vote
0answers
43 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
1
vote
1answer
36 views

Krull dimension and zero divisors of $k[x,y,z]/(x^ay,x^bz)$

I found the primary decomposition of $(0)$ in the ring $k[x,y,z]/(x^ay,x^bz)$, where $a\geq b \geq 1$, $k$ is alg. closed, to be $(x^b) \cap (x^a,z) \cap (y,z)$ (is this correct?). Now I am now ...
1
vote
1answer
31 views

Presentation of a local complete intersection

What is the simplest example of a local (noetherian) complete intersection ring $R$ that can not be presented as $R=S/I$, where $S$ is a regular local ring and $I$ is an ideal generated by a regular ...
1
vote
1answer
34 views

closed and open subscheme of affine scheme

Let $X=Spec(A)$ be a noetherian affine scheme. Let $I_1, \ldots, I_n$ be ideals of $A$ such that $I_i + I_j = 1$ for all $i \neq j$. Define $X_i = Spec(A/I_i)$ so that X is the disjoint union of the ...
0
votes
1answer
49 views

A question about varieties (proposition from Miles Reid undergraduate comm algebra)

In Miles Reid, undergraduate commutative algebra, I read the following: "Suppose that $k$ is an algebraically closed field and that $A=k[x_1,...,x_n]$ is a finitely generated $k$-algebra of form ...