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

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1answer
131 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
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1answer
63 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} ...
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0answers
47 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 $\mathcal{S}$ ...
4
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1answer
224 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
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1answer
70 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 ...
1
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1answer
58 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 ...
9
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2answers
276 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 $\mathrm{Spec}(\mathbb{Z})$?
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2answers
67 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 ...
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1answer
65 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
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2answers
91 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 ...
3
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1answer
70 views

The inverse of a fractional ideal is a fractional ideal

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 ...
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1answer
34 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
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1answer
48 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
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2answers
116 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). ...
2
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1answer
201 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 ...
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0answers
98 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
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1answer
73 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
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1answer
93 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
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1answer
82 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:
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1answer
109 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} ...
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2answers
75 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
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1answer
62 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
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1answer
76 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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1answer
40 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), \ ...
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0answers
51 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 ...
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0answers
68 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
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1answer
61 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 ...
2
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2answers
52 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
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1answer
116 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
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1answer
44 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 ...
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0answers
69 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 ...
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1answer
45 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 ...
2
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1answer
69 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 ...
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0answers
43 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 ...
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0answers
52 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
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1answer
89 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
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1answer
64 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 ...
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1answer
115 views

Example of Gorenstein local ring of dimension 1

The ring $k[[x,y]]/(xy)$ is Gorenstein. Why?
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1answer
60 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 = ...
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0answers
67 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
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1answer
62 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
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1answer
41 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
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1answer
65 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 ...
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1answer
75 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 ...
3
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1answer
123 views

When a holomorphy ring is a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
3
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1answer
53 views

Dimension of irreducible variety

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?
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0answers
43 views

Relation about prime ideals in $B$ and invariant subring $B^G$

Suppose $B$ is a commutative ring, $G$ is a finite group acting on $B$, $A=B^G$ is the invariant subring. Suppose $P$ is a prime ideal in $A$, $Q_1,...,Q_s$ are all the prime ideals in $B$ such that ...
1
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1answer
596 views

Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
1
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0answers
52 views

When is a holomorphy ring a PID? [duplicate]

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
1
vote
1answer
68 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...