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

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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
52 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
55 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 ...
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91 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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1answer
63 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
59 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 $\{0\}$ ?
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197 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 ...
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91 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 ...
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0answers
116 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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159 views

Problem on the number of generators of some ideals in $k[x,y,z]$ [closed]

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
106 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 ...
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1answer
115 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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2answers
168 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
42 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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2answers
127 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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72 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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75 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
106 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 ...
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1answer
109 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
96 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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40 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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1answer
121 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
95 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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71 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 ...
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41 views

Deducing a presentation for a complete intersection

Today, while reading some articles, I had this doubt trying to justify a passage: Hypothesis Suppose $O_K$ is some complete discrete valuation ring (it is the ring of integers of some complete field ...
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143 views

Computing the Length of a finite length module.

How we can compute the length (length of a composition series) of the Artinian local ring $R=K[x,y]/(x^3,y^3)$ ? Does the following chain is a saturated chain of ideals of $K[x,y]$ ? ...
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1answer
101 views

Is Serre's $S_1$ condition equivalent to having no embedded primes?

Today I tried to prove that if a Noetherian ring $A$ satisfies Serre's $R_0$ and $S_1$ conditions, then $A$ is reduced. Now we recall that $R_0$ means the localization at any minimal prime is a field ...
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1answer
136 views

The kernel of $R \to A \otimes_R B$ is nil

Let $R \to A$ and $R \to B$ be two homomorphisms of commutative rings whose kernels are nil (i.e. consist only of nilpotent elements). Then the kernel of $R \to A \otimes_R B$ is also nil. See ...
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question about gradation of a ring

I was reading Mumford's 'Red book on varieties and schemes', when I came across the following paragraph: I am confused about meaning of the phrase "We let $k(X)$ be the zeroth graded piece of the ...
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4answers
393 views

A finite dimensional algebra over a field has only finitely many prime ideals and all of them are maximal

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
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109 views

Computing generators of the positive component of a graded ring

Let $R$ be a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$ given by finitely many generators, and let $\lambda$ be a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. This defines a ...
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54 views

An example of a henselian valuation of rank 2.

I need to know simple examples of valuations of rank bigger than one. Please help me to concrete some examples of valuations of rank bigger than one with their valuation rings (specially henselian ...
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1answer
77 views

Are locally integral extensions integral? [closed]

Let $R$ be a commutative ring with unit and let $A \subseteq B$ be commutative $R$-algebras. Suppose $B_P$ is integral over $A_P$ for all primes $P$ of $R$. Is $B$ then necessarily integral over ...
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1answer
54 views

Are there any integral domains in which no nonzero prime ideal is finitely generated?

Are there any integral domains in which no nonzero prime ideal is finitely generated? (Other than fields, of course, where the condition is vacuously satisfied.) I asked a similar question the ...
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1answer
240 views

Tensor product of injective ring homomorphisms

What is an example of two injective homomorphisms $R \to A$, $R \to B$ of commutative rings such that $R \to A \otimes_R B$ is not injective? Of course neither $R \to A$ nor $R \to B$ can be flat in ...
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80 views

Extending a lemma about Castelnouvo-Mumford regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. There is a lemma: It's from "Castelnuovo-Mumford regularity postulation number and relation types" by Markus Brodmann and ...
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1answer
94 views

Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated? I feel like the example (or proof of impossibility) ought to be obvious, but I'm not seeing it.
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1answer
70 views

question about $Spec(A)$ in Atiyah's book Introduction to Commutative Algebra

Let $A$ be a ring and $X=spec(A)$, the prime spectrum of $A$. Prove that $X$ is quasi-compact. Definition of quasi compact: each open covering of $X$ has a finite subcovering of $X$. It is ...
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41 views

If $Q$ is a prime ideal of $R[x]$ then $QF[x]\cap R[x]=Q$

I'm filling the gaps in a proof and I'm stuck in this part: Suppose $R$ is a UFD and $Q$ is a prime ideal of $R[x]$, if $F$ is the quotient field of $R$ and $R\cap Q=\{0\}$, then $QF[x]\cap ...
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1answer
68 views

Subtlety in Correspondence Theorem for Rings

I have something of a subtle question about the correspondence theorem for rings. The theorem is typically stated like this: Let $A$ be a ring, and $I$ an ideal of $A$. There is a $1-1$, ...
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47 views

Maximal $R$-sequences in ideals

If $\alpha_1,...,\alpha_s$ is a maximal $R$-sequence in an ideal $I$ ($R$ is commutative with unity), is this always true that $I⊆P$, where $P\in\operatorname{Ass} (\alpha_1,...,\alpha_s)$? In case ...
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A question about the size of reduced Groebner basis

Let $I=(f,g,h)$ be an ideal in the polynomial ring $k[x,y,z]$ with $LT(f)>LT(g)>LT(h)$ in the lexorder, and $I$ is "reduced" in the sense that $LT(g)\nmid LT(f),LT(h)\nmid LT(g),LT(h)\nmid ...
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141 views

What are some examples of coolrings that cannot be expressed in the form $R[X]$?

Let $K$ denote a field. Then the polynomial ring $K[x]$ has the property that the sum of two units is either a unit, or zero. I'll bet there's heaps of other examples, though. So let a coolring be a ...
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1answer
50 views

If $R\rightarrow S$ is faithfully flat then show that it is pure, and reference for purity

I was reading about $F$-purity and $F$-splittings, when I came across then following statement which I can't prove: Definition: Let $R$ be a commutative ring with identity, and $M,N$ be $R$-modules. ...
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1answer
126 views

Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
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164 views

Thinking About Fractional Ideals Geometrically

So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This ...
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88 views

Castelnouvo-Mumford Regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. As u see in 2 pics below, Lemma 3.3. $(A,\mathfrak{m})$ is a Noetherian ring local, $\dim(A)=1$; $\mathfrak{q}=(x)$ is a ...
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1answer
77 views

How to define a smooth subvariety as the vanishing of local coordinates

I keep stumbling upon this fact, and would like to see or get an idea for the proof: An ideal of a smooth subvariety at a point of a smooth variety can be generated by a subset of a suitably chosen ...
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1answer
121 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
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1answer
139 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 6.4.17 (b) [closed]

Let k be an infinite field, $S=k[x_{11},x_{12},x_{21},x_{22}]/(x_{11}x_{22}-x_{12}x_{21})$. Let $p=(x_{11},x_{12})$, $q=(x_{21},x_{22})$. Show that (i) $p$ and $q$ are prime ideals in $S$ ...