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

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1answer
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$\hom_R(A,B)$ is finitely generated if $R$ is noetherian [duplicate]

This is part of an exercise I'm doing, from Rotman Introduction to homological algebra. Let $R$ be a commutative ring, and let $A$ and $B$ be finitely generated $R$-modules. Then if $R$ is ...
2
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1answer
119 views

Surjectivity between Local Noetherian Rings

My friend and I were working through old preliminary exams. This problem is the fifth one on this list, but we were unable to come up with a solution. Let $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ ...
1
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2answers
198 views

Maximal ideals of a finite dimensional algebra over a field

Let $\mathfrak o$ be a Dedekind domain, $K$ its field of fractions, $L$ a finite separable field extension of $K$, $\mathfrak O$ the integral closure of $\mathfrak o$ in $L$. Let $\mathfrak p$ be a ...
1
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1answer
190 views

“Local free”ness in vector bundles and projective modules

Swan's Theorem tells us that (real) vector bundles on $X$ are the same as finitely-generated projective modules over $C(X)$ (continuous $\mathbb{R}$-valued functions on $X$. And vector bundles are ...
1
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0answers
33 views

Looking for an example for a very particular kind of cone

I am looking for a cone $X\subseteq\mathbb A^n_{\mathbb C}$ (defined by homogeneous equations) and an irreducible homogeneous polynomial $f$ in $n$ variables such that $U := D(f)\cap X = X_f = \{ x\in ...
1
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1answer
65 views

Prove the strong Nullstellensatz from these two conditions

It is an exercise. Let $R=k[x_1,\dots,x_n]$ where $k$ is an algebraically closed field. Assuming that (1) $R$ is Noetherian, and (2) the maximal ideas of $R$ are precisely the ideals of ...
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0answers
81 views

Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
2
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1answer
119 views

Localization at primes and local ring

I'm now studying localization. When $R$ is a commutative ring, I know that localization at a prime is a local ring. I have a question. Let $\mathfrak p_{1},\dots,\mathfrak p_{n}$ be primes of $R$ ...
2
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1answer
360 views

Every prime ideal of a finitely generated $\mathbb{R}$-algebra is an intersection of maximal ideals?

Why must every prime ideal of a finitely generated $\mathbb{R}$-algebra (e.g. $\mathbb{R}[X_1,X_2]$) be the intersection of the maximal ideals containing it? This doesn't follow from the version ...
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3answers
63 views

$a^n = 0 \implies a \in P$ (where $P$ is a prime ideal)

Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!
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1answer
44 views

Hilbert's Basis Theorem question

If $F$ is a field and $R = F[t_1, t_2, ... t_k]$ and $Y$ is a set of polynomials in $k$ variables over $F$ then by Hilbert's basis theorem apparently $YR = \sum\limits_{i=1}^m f_i R$ for some ...
3
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2answers
134 views

If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated.

I'm answering this question: Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever ...
3
votes
1answer
138 views

$\mathcal{O}_X(D)$ is invertible implies $D$ is locally principal

With words added for context, 14.2.G of Ravil Vakil's notes asks Suppose $X$ is an integral, normal and Noetherian scheme, and $D$ a Weil divisor. Let $\mathcal{O}_X(D)$ be the quasicoherent sheaf ...
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0answers
54 views

When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
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2answers
52 views

There are elements in $\mathbb Z [t]$ such that this sum is 1

In order to solve a problem I'm facing I want to prove that there are $f_1,f_2,f_3$ elements in $\mathbb Z[t]$ such that $f_1(t)\cdot(4t-4)+f_2(t)\cdot(5t)+f_3(t)\cdot(t^2-17)=1$. In another words, I ...
2
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1answer
105 views

Commutative algebra with a geometric flavor

Does anybody know where can I find a book with topics similar to the ones in Atiyah's Introduction to commutative algebra, but with some sort of geometric motivation? Thanks!
2
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1answer
107 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
2
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1answer
42 views

Question on finitely presented algebra

Suppose $S$ is a finitely presented $R$-algebra. If $g:R[x_1, \ldots, x_n] \to S$ is surjective, then $\ker(g)$ is finitely generated. We can write $S$ as $R[y_1, \ldots, y_m]/(f_1,\ldots,f_t)$ ...
1
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1answer
62 views

Associated primes of quotient module

Let $R$ be a Noetherian local ring of Krull dimension $d$, $M$ a finitely generated module over $R$. Suppose $\dim M=d$ and $K$ is a submodule of $M$ maximal with respect to the property that $\dim ...
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0answers
38 views

Any counterexample for inverse limit functor not to be right exact [duplicate]

We know that inverse limit is a "left" exact functor on the category of modules in the sense that whenever $r:(A_i,α_j^i )→(B_i,β_j^i )$ and $s:(B_i,β_j^i )→(C_i,γ_j^i )$ are transformations of ...
1
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1answer
40 views

A nonexample of a prime submodule?

Let $M$ be an $A$-module. A submodule $P\subset M$ is called a prime submodule if it is proper and $am\in P$ implies $aM\subset P$ or $m\in P$. It is easy to see that if $P\subset M$ is a prime ...
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1answer
33 views

Example of non finitely-generated $R$-Module $K$ so that $K/PK$ is finitely generated

I am studying about this and questioning some problems. Suppose $F$ is a field and $R=F[x]_{(x)}$, the localisation of $F[x]$ at prime ideal $P=(x)$. I am trying to find a non finitely-generated ...
2
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1answer
55 views

Computing Poincare Series of a particular graded Algebra

The definition for the Poincare series of a graded algebra over a field $k$, $A=A_0 \oplus A_1 \oplus A_2 \oplus \cdot \cdot \cdot$ , is $P(A)= \sum_{i \geq 0} \text{dim}_k(A_i)t^i$. I'm trying to ...
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1answer
104 views

Errata with Eisenbud's Lemma “Symmetry of Diagonalization” proof.

In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like this: Let R be a commutative ring, $M$ an $R$-module, $S(M)$ ...
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1answer
42 views

Integral closure of a subring that is a polynomial ring over an algebraically closed field. [closed]

Let $K$ be an algebraically closed field that is a subring of an integral domain $D$. Assume $D$ contains an element $d$ that is transcendental over $K$. Also assume that $D$ is integral over ...
4
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1answer
167 views

Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
1
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1answer
59 views

$\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$

In a proof (proof of theorem 4.3.36 in Liu's book) I need the equality $\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$. The hypothesis of the theorem are the following: $Y$ ...
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0answers
89 views

Rings having the same characters but not isomorphic.

I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
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1answer
58 views

Ideal of an integral domain all of whose exterior powers are nonzero.

I want to find an integral domain $R$ with ideal $I$ (considered as an $R$-module) such that $\bigwedge^k I\neq 0$ for all nonnegative integers $k$. Dummit and Foote gave the example of $R=\mathbb ...
2
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1answer
46 views

Does there always exist finitely presented submodules?

Suppose $M\neq 0$ is an $A$-module where $A$ is a commutative ring with $1$. Is it always possible to find a finitely presented submodule $N\neq 0$ of $M$? It is not interesting when the ring ...
3
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2answers
175 views

Local ring of Krull dimension zero

In commutative rings text books it is usually asked to prove that as long as $(R,m)$ is a Noetherian local ring, the following are equivalent: (i) $m^n=m^{n+1}$ for some integer $n$; (ii) ...
2
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2answers
109 views

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$?

What can be said about the relation between $\mathbb{Z}_p$ and $\mathbb{Z}_{(p)}$? This is a question in Hungerford. I understand what both are, $\mathbb{Z}_p = \mathbb{Z}/(p)$ is a finite field and ...
1
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1answer
64 views

What is $R$-algebra and do I need to understand $R$-modules for it?

I was given the following definition of $R$-algebra: Let $R$ be a commutative ring. An $R$-algebra is a ring $A$ (with $1$) together with a ring homomorphism $f : R \to A$ such that ...
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1answer
81 views

Exact sequence induced by decomposition of a variety

Let $X=Y_1\cup Y_2$. Denote $I:=\mathcal{I}(X)$, $I_1:=\mathcal{I}(Y_1)$, $I_2:=\mathcal{I}(Y_2)$. Then we have $I=I_1\cap I_2$. In this question, the answer suggests that there is an exact ...
4
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0answers
132 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
0
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1answer
133 views

How to prove Ass$(R/Q)=\{P\}$ if and only if $Q$ is $P$-primary when $R$ is Noetherian? [duplicate]

Let $R$ be a Noetherian ring, $P$ be a prime ideal, and $Q$ an ideal of $R$. How to prove that $$ \text{Ass}(R/Q)=\{P\} $$ if and only if $Q$ is $P$-primary? Update In fact, I have proved that if ...
3
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2answers
223 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
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1answer
65 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
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2answers
38 views

induced sequence exact

If $D$ is a multiplicatively closed subset of $R$. I'm trying to come up with an example where $$0\to L \to M \to N \to 0$$ is not exact, but the induced sequence $$0 \to D^{-1}L \to D^{-1}M \to ...
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1answer
66 views

Is a complete Noetherian local ring a homomorphic image of a Gorenstein local ring of the same dimension?

From the proof of Theorem 29.4 in Matsumura's book Commutative Ring Theory we can see that a complete Noetherian local ring is a homomorphic image of a regular local ring, but how can we prove that a ...
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3answers
83 views

$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.

How to prove : $A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal. Any suggestions ?
3
votes
1answer
105 views

Integral Domain with exactly two Prime Ideals

I am not looking for someone to give me an explicit example. I want to work this out myself if possible. Trying to learn schemes by reading The Geometry of Schemes by Eisenbud and Harris. Problem I-5 ...
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1answer
82 views

Retraction for rings?

For abelian groups, the existence of left inverse or right inverse of a homomorphism can be characterized by looking at whether the image or kernel splits the group. Is there an analogous ...
0
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1answer
53 views

Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
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1answer
77 views

Injective modules in a short exact sequence

Let $0→A→B→C→0$ be an exact sequence in the category of $R$ modules, where $R$ is commutative with $1$, and $B$ be injective. In a text book it is said that all three modules are injective, or the ...
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2answers
93 views

Injective resolution for an integral domain [closed]

How could one write an injective resolution for an arbitrary commutative integral domain $R$? Thanks in advance!
3
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4answers
197 views

Finding generators for an ideal of $\Bbb{Z}[x]$

We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime ...
3
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1answer
255 views

Matsumura, Exercise 18.8: Cohen-Macaulay and (not) Gorenstein [duplicate]

I need an answer to the exercise 18.8 of Matsumura's book:" Commutative ring theory", and generate an algorithm if possible. Let $k$ be a field and $t$ an indeterminate. Consider the subring $A = ...
2
votes
1answer
107 views

If $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody: if $R$ is a noetherian local ring, then every 2-generated ideal has ...
1
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1answer
141 views

Example of Noetherian ring over which the Euclidean algorithm is not valid.

As stated in the question, I am looking for a Noetherian ring over which the Euclidean algorithm is not valid. I am trying to construct non-trivial examples of Noetherian rings. Thank you.