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

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Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c)

The following question is from the book: Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c). Let R be a ring, I a finite generated ideal and M an R-module. Let R$_{\infty}$ be a polynomial ...
1
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1answer
30 views

Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to ...
3
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2answers
55 views

Localization at finitely many minimal prime ideals

Let $A$ be a commutative ring with finitely many minimal prime ideals $\{p_1,\dots,p_n\}$. Let $A_{p_1,\dots,p_n}$ be the localization of $A$ away from the minimal primes, i.e. $S^{-1}A$ where $S = ...
2
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1answer
42 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
1
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1answer
34 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
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1answer
59 views

Is $\mathbb Z\oplus \mathbb Q$ a flat $\mathbb Z$-module? [on hold]

Can someone please explain why $\mathbb Z\oplus \mathbb Q$ is flat or not?
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1answer
51 views

What are good references for the theory of Cohen-Macaulay rings?

I am studying Cohen-Macaulay Rings from the book Cohen-Macaulay rings by W. Bruns. Please tell me some reference book/notes on Cohen-Macaulay rings theory.
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1answer
59 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
4
votes
3answers
78 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
2
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0answers
35 views

Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
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0answers
52 views

Is the maximal ideal of a localization at a prime ideal principal?

Let $X$ be a closed subvariety of $\mathbf P^{n}_{k}$ which is nonsingular in codimension one. Let $Y$ be a subvariety of $X$ of codimension one, let $\eta$ be its generic point. First question: is ...
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2answers
160 views

Is Pythagoras the only relation to hold between $\cos$ and $\sin$?

Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$. (Vague) Question. Is this the only relationship between the functions $\cos$ and $\sin$? More precisely: Let ...
0
votes
1answer
61 views

Radical ideals of $\mathbb{Z}$?

I am having trouble with classification of the radical ideals of $\mathbb{Z}$. We know that for a commutative ring $R$ with an ideal $I$, the radical of $I$ is defined (and denoted as $\sqrt{I}$) as ...
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0answers
84 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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0answers
30 views

Relation between elements of a ring and their annihilators

let $(R.m)$ be a local ring and $x,y$ two elements of $R$ and for ideal $I$ of $R$, we have $x$ is in $I$, $ann(x)=ann(y)$ and $x$ is uniqu minimal ideal of $R$, is there any conditions that implies, ...
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1answer
48 views

A property of minimal prime ideal

Let $R$ commutative ring with unity, $S\subseteq R$ subring, $p$ minimal prime ideal of $S$. Show there exists a minimal prime ideal $q$ in $R$ with the property that the contraction $q^c=q\cap S=p$. ...
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1answer
44 views

A question about projective modules.

Suppose that we have a commutative ring $R$ with an idempotent $e$, and $M$ an $R$-module such that $Me$ is $Re$-projective. I am interested to know under which conditions this implies that $M$ is ...
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0answers
23 views

Frobenius splitting from viewpoint of commutative algebra

First I define two terms: Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...
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0answers
35 views

Gorenstein ring and projective module

I am new to this topic and would appreciate little explanation. Def: A commutative, unital ring $A$ is a cubic ring if $A$ is a free $\mathbb{Z}$-module of rank $3$. Def : A cubic ring $A$ is ...
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0answers
18 views

a question on equivalence classes of balanced fractional ideals and Dedekind domain

Let $R$ be a commutative ring, and let $K=R\otimes \mathbb{Q}$. Def.1) We say that a pair of fractional ideals $(I, I')$ in $K$ is balanced if $II'\subseteq R$ and $N(I)N(I')=1$. Def.2) Two ...
2
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1answer
45 views

Preimage of maximal ideal is maximal [duplicate]

I've just started a commutative algebra course and I'm stuck on the very first homework problem: Let $A \not= \{0\}$ be a commutative ring. Let $\Phi : A \longrightarrow B$ be a ring homomorphism ...
3
votes
1answer
48 views

Minimal Number of Generators

In commutative algebra, for a module $M$ over a (possibly unital) commutative ring $R$, when is the number $\mu_R(M)$ well-defined? For example, if $R$ is a local ring, then (by Nakayama Lemma and ...
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1answer
54 views

Intersection of two polynomial ideals

In the 4-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
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1answer
35 views

Relation between $\text{Hom}_{\mathsf{Alg}_{\mathbb{R}}}(\mathcal{C}^\infty(M),A) $ and $ X \otimes_\mathbb{R} A$?

This question is a little bit of a shot in the dark, but maybe someone stumbled over it before... Let $M$ be a (simply connected) smooth manifold modelled on a locally convex space $X$ over ...
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1answer
37 views

If the factor of a finitely generated module is free then submodule is also finitely generated

All rings are commutative, associative and with 1. Consider short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ of $R$-modules. How to show that if $M$ is finitely ...
3
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0answers
74 views

preservation of localness among certain Krull domains

Let $R$ be a local Krull domain, and let $\mathfrak p$ be a height one prime ideal whose class in the divisor class group is non-torsion. (That is, $\mathfrak p^{(n)}$ is non-principal for all $n$.) ...
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2answers
115 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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1answer
21 views

Length of polynomial ring modulo a homogeneous regular sequence

Proposition: Let $k$ be a field and $R=k[x_1,\dots,x_n]$ the polynomial ring with $x_i$ having degree $1$. Let $f_1,\dots,f_n$ be homogeneous elements such that $\deg(f_i)=s_i >0$ and they form ...
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1answer
29 views

An equivalent condition for zero dimensional Noetherian local rings

Let $(A,m)$ be a Noetherian local ring. Why "$A$ is zero dimensional if and only if a power of $m$ is $\{0\}$"?
8
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1answer
161 views

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
1
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1answer
78 views

non-principal height one primes of a particular hypersurface

I was reading about divisor class groups, and I was wondering the following. Let $R=\mathbb{C}[X,Y,Z,W]/(XZ-YW)$, and let $x,y,z,w$ be the images of $X,Y,Z,W$ in $R$, respectively. Is there a way ...
2
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0answers
73 views

The greatest common divisor of homogeneous polynomials

Let a matrix $$M=\begin{pmatrix} a_{01}&a_{02}&a_{03}\\a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$$ with $a_{ij}\in k[x,y,z]$ ...
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1answer
39 views

Problem on the number of generators

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements Here the ...
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1answer
72 views

Intersection of all associated primes

Given $(R,m)$, a Noetherian local ring, and $M$ a nonzero $R$-module. I was wondering if there is a way to describe the elements of $\displaystyle\bigcap_{P\in Ass_RM} P$. In particular, when $M$ is ...
2
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1answer
32 views

Localization of a finitely generated module is trivial iff its annihilator is nontrivial

I have a problem on Atiyah and MacDonald's commutative algebra book, the exercise 3.1: Let $S$ be a multiplicatively closed subset of a ring $A$ and $M$ a finitely generated $A$ - module. ...
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0answers
22 views

Kernel of homomorphism $A[X] \to B$ between integral domains [duplicate]

Let $A \leq B$ be integral domains, where $A$ is integrally closed and $B/A$ is an integral ring extension. Let further $\varphi : A[X] \to B$ be some homomorphism of $A$-algebras. Is the kernel ...
2
votes
2answers
84 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are ...
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0answers
22 views

Extension of graded algebra by a homogeneous ideal

If an algebra is graded by the group $G$: $A=\bigoplus\limits_{d \in G} A_d$ and contains a homogeneous ideal $I \subset A$, then we have the quotient $B:=A/I$ and canonical epimorphism $\nu:A ...
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1answer
49 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
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1answer
38 views

Nilpotent elements in commutative rings

Let $A$ be a commutative ring, $a, a+b \in A$ are nilpotent. Does this imply that $b$ is nilpotent?
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1answer
62 views

The set of all $p \in \mathbb{C}[x]$ that can be expressed using only one occurrence of $x$.

Let $X$ denote the least subset of $\mathbb{C}[x]$ subject to the following constraints. $x \in X$. $p \in X \rightarrow ap \in X,$ for all $a \in \mathbb{C}$. $p \in X \rightarrow p+a \in X,$ for ...
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1answer
59 views

Does there exist a UFD having only finitely many irreducibles?

Does there exist a UFD (which is not a field) having only finitely many irreducible elements? Definition of a UFD is: $R$ is an integral domain ($R$ is a commutative ring having unity and no ...
3
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1answer
69 views

Plane curves isomorphic to the affine line

Let $C$ be a plane curve parametrized by $x=f(t),y=g(t)$ where $f(t),g(t)\in k[t]$. We can easily see that the coordinate ring of $C$ is isomorphic to $k[f(t),g(t)]\subset k[t]$. So $C$ is isomorphic ...
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1answer
38 views

Difference between PID and principal ideal ring

All rings are commutative, associative and with 1. Wikipedia states that the difference between PID and Principal Ideal Ring is that the former has to be integral domain while the latter does not. ...
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0answers
35 views

Interpretation of $\Omega_{A/k} \simeq A \otimes_k I/I^2$ for affine group schemes

I'm learning some group scheme stuff and there's the following result: If $A$ is Hopf $k$- algebra, then $\Omega_{A/k} \simeq A \otimes_k I/I^2$, where $I$ is the augmentation ideal. I know the ...
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0answers
30 views

A question about locally free differential sheaf and regular local ring

Let $B$ be a local ring containing a field $k$ isomorphic to its residue field. Assume furthermore that $B$ is a localisation of a finitely generated $k$-algebra. Then $Ω_{B/k}$ is a free $B$-module ...
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1answer
73 views

Atiyah-MacDonald, Problem 6 of Chapter 1

I was trying to solve the following problem from "Introduction to Commutative Algebra" by Atiyah and MacDonald. (It is Problem 6 of Chapter 1.) While trying to solve the problem, I am facing trouble ...
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1answer
49 views

A nonfree module which is locally free

The general context is trying to understand the Picard groups of various schemes, but this question focuses on affine schemes. Let $X=Spec A$ an affine scheme. What conditions does $A$ need to ...
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0answers
28 views

Same number of generators and relations in a complete intersection, when?

I make this question a bit more general because i think as i put it, it will have no answer because there are too many maybe irrelevant details: Given $B$ an $A$-algebra, local, of finite type (that ...
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0answers
57 views

Local complete intersection ring

Suppose $R$ is a local Noetherian complete intersection ring that is a finite $A$-algebra, where $A$ is a DVR. If the module of differentials of $R$ is free as an $R/\mathfrak a$-module for some ...