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

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Jacobson radical of formal power series ring

If $f=\sum_{i=0}^{\infty} a_i x^i \in R[[x]]$, let $\mathfrak{R}$ denote the Jacobson radical of a ring. I wish to show that $f\in\mathfrak{R}(R[[x]])\iff a_0\in\mathfrak{R}(R)$. I have already proved ...
4
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0answers
52 views

What are “generalized bases” really called, and where can I learn more?

(Notation: $f \diamond g$ means the composite $g \circ f$.) The following situation occurs frequently: We have an $\mathbb{R}$-algebra $A$, together with a distinguished set $I$ (the "indexing set"),...
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1answer
57 views

Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
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1answer
54 views

If $A$ is an integral domain with a finite number of primes then $Q(A)=A_a$ for some $a \in A$ [closed]

If $A$ is an integral domain with a finite number of prime ideals is it possible to get the field of fractions localizing only by a set $\{a^k\}$?
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42 views

Global dimension of power series ring $k[[x_{1}, \cdots, x_{n}]]$

Let $R$ be the power series ring $k[[x_{1}, \cdots, x_{n}]]$ over a field $k$. Notice that $R$ is a noetherian local ring with residue field $k$. Show that $gl. \dim(R)=pd_{R}(k)=n$. By First Change ...
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1answer
32 views

Total quotient ring of $\mathbb Z_{2^n}$ [closed]

I want to characterize the total classical quotient rings of the (commutative) rings $R=\mathbb Z_4$, $R=\mathbb Z_8$ or any $R=\mathbb Z_{2^n}$. In fact, if we get $S$ to be the regular elements ...
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2answers
65 views

If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ [duplicate]

If $I$ and $J$ are ideals in a ring $R$ with 1 which are co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$ Work done: Should I proceed using Zorn'...
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1answer
74 views

Is the coordinate ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ or the ring $\mathbb{C}[t,t^{-1}]$ unique factorization domains?

I wonder if either the coordinate ring $A(X):=\mathbb{C}[x,y]/(x^2+y^2-1)$ or the polynomial ring $\mathbb{C}[t,t^{-1}]$ are unique factorization domains? I know that those are isomorphic, so the ...
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4answers
186 views

Diagonal morphism of regular variety is a regular embedding

Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding ...
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1answer
84 views

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$?

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$? I tried to decompose $$(z^4,x^2+y^2+z^2-1,xy)=(z^4,x^2+y^2+z^2-1,x)\cap(z^4,x^2+y^2+z^2-1,y)=(z^4,x^2+z^2-1,y)\cap(z^4,y^2+...
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2answers
48 views

Noether Normalization of $\mathbb{C}$-algebra $\mathbb C[x,y,z]/(xy+z^2,x^2y−xy^3+z^4−1)$.

I'm trying to find a Noether normalization of the $\mathbb{C}$-algebra $$\mathbb C[x,y,z]/(xy+z^2,x^2y−xy^3+z^4−1).$$ The proof of Noether Normalization Theorem is not constructive. I choose $x = {x_1}...
1
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1answer
34 views

What should I call an “injective” algebra?

Given rings $A,B$, we say that $B$ is an $A$-algebra if there is a ring homomorphism $f:A\rightarrow B$. This homomorphism give the structure of the algebra. Various properties of algebras can either ...
2
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1answer
51 views

Isomorphism which involves $\mathbb Z_p[[T]] \otimes \mathbb Q_p$

Why should $\mathbb Z_p[[T]] \otimes_{\mathbb Z_p} \mathbb Q_p$ be isomorphic to the bounded sequences with values in $\mathbb Q_p$? The fact is that the tensor product is on $\mathbb Z_p$, so it is ...
2
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1answer
53 views

$A$ local Noetherian ring with principal maximal ideal implies PIR?

Suppose that $A$ is a local Noetherian ring with principal maximal ideal. Can we prove that every ideal of $A$ is principal? I tried to exploit the Noetherian property on the set of non-principal ...
5
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1answer
86 views

Hilbert polynomial for a dimension zero projective variety by taking an affine chart

I am looking at exercise 12.21 from Gathmann's notes on algebraic geometry. I am given a homogeneous ideal $$I \unlhd k[x, y, z] $$ with a dimension $0$ projective locus. WLOG, we assume that this ...
1
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1answer
39 views

Stabilization of colon ideals of a decomposable ideal

Let $\mathfrak a$ be a decomposable ideal in a (commutative ring with unity) $A$, let $\Sigma$ be an isolated set of prime ideals belonging to $\mathfrak a$, and let $\mathfrak q_\Sigma$ be the ...
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0answers
46 views

Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
2
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1answer
210 views

Auslander-Buchsbaum formula - proof

I have a final exam in commutative algebra and Auslander-Buchsbaum formula is one of the theorems that we have to self-study for the exam. Unfortunately, our course did not cover minimal resolutions ...
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0answers
34 views

Locally finitely generated ideal imply finitely generated

Let $R$ be an integral domain and $I$ a non-zero ideal of $R$. Let $0\neq a \in I$ and suppose that $a$ is contained in only finitely many maximal ideals $Q_1,...,Q_n$. Further suppose that each $IR_{...
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1answer
84 views

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$?

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$? I understand that the ideals are primary and also that one has $$(x,y)\cap(x,z)\cap(x,y,z)^2=(x,y)(x,z).$$ But I ...
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1answer
117 views

Are $C(\mathbb{R})$ and $D(\mathbb{R})$ isomorphic or not?

Let's denote ring of all continuous functions and differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ by $C(\mathbb{R})$ and $D(\mathbb{R})$, respectively. I want to know whether these rings ...
2
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1answer
40 views

Support of quotient sheaf of ideal sheaves with same support

I'm not very sure about this argument. Let $\mathscr{I},\mathscr{J}$ two ideal sheaves (you can think about ideal sheaves over a projective variety or even the projective space itself) and assume that ...
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1answer
28 views

Proposition about rings of fractions

This is taken from Atiyah-Macdonald's Commutative algebra book page 41. Someone please explain to me what is the meaning of "$a$ meets $S$". This is the first time I'm seeing this in the book.
2
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1answer
71 views

$A^\times/k^\times$ is a free $\mathbb{Z}$-module of rank of at most $r - 1$

Consider an algebraically closed field $k$, a finite extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, the integral closure $A'$ of $k[1/T]$ in $K$, and the integral closure $A''$ of ...
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1answer
129 views

Is the proof of Auslander-Buchsbaum formula in Rotman's Advanced Modern Algebra wrong?

Suppose that $(R,m,k)$ is a commutative, local, Noetherian ring. If $F$ is a free $R$-module then $\operatorname{Ext}^i_R(k,F)=0$ for $i\geq0$. This statement is part of a proof for the Auslander-...
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1answer
38 views

Leading term ideal irreducible iff the ideal is irreducible?

Let $\mathbb{K}$ be a field. Given an ideal $I \subset \mathbb{K}[x_1,\dots, x_n]$ and a monomial order we can consider the ideal $LT(I) = (lt(f) \ | \ f\in I )$, where $lt(f)$ denotes the leading ...
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1answer
63 views

Showing $K = I \otimes_R K$, where $K$ is the field of fractions of $R$.

A question in a previous Commutative Algebra Exam reads: Problem. Let $R$ denote an integral domain, $K$ its field of fractions. Let $I$ denote a non-zero ideal of $R$. Show that $K=I \...
2
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2answers
56 views

Basic question about ideals in a polynomial ring.

I would be very grateful if someone would verify or refute the following solution. Many thanks! Q) Find infinitely many distinct ideals of $\mathbb{C}[X,Y]$ which contain the principal ideal $(X^3-...
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1answer
33 views

Proof that primary submodules of $R$ are primary ideals of $R$

I want to prove this: Let $R$ be a commutative ring with identity. If $Q$ is a primary submodule of $R$ (as an $R$-module), then $Q$ is a primary ideal. $Q$ is a primary submodule of $R$ if $r \in R$...
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0answers
68 views

Do we have that $y\in R\setminus R^\times$, such that $y^n\mid x$ for every $n\in \mathbb N\implies x=0$?

Is it true that if $R$ is an integral domain with $1$, and for $x\in R$ there exists $y\in R\setminus R^\times$ such that $y^n$ divides $x$ for every $n\in \mathbb{N}$, then $x=0$? I think this is ...
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0answers
41 views

Localization and the universal property

I've been trying to get to grips with the universal property and was looking at the localization of a commutative ring $A$ at some multiplicative set $S$, denoted $S^{-1}A$, to get an idea of what the ...
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1answer
47 views

Localization of $\mathbb{Z}/p^k\mathbb{Z}$ at $S=\begin{Bmatrix}b^n : n\in \mathbb{N}\end{Bmatrix}$

Could I say that $\left(\mathbb{Z}/p^k\mathbb{Z}\right)_{b}$, namely the localization at $S=\begin{Bmatrix}b^n : n\in \mathbb{N}\end{Bmatrix}$ when $(b,p)=1$, is equal to $\mathbb{Z}/p^k\mathbb{Z}$ ...
5
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2answers
160 views

Relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring for dummies?

As the question title suggests, what is an explanation for dummies of the relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring?
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34 views

If $A$ is a local ring, is it always true that $(A/Q)_{\mathfrak{m}}\cong A/Q$?

$A$ is a local ring with $\mathfrak{m}$ its maximal ideal and $Q\subset A$ is an ideal of $A$. I thought that $$(A/Q)_{\mathfrak{m}}\cong A/Q\otimes_{A} A_{\mathfrak{m}}\cong A/Q\otimes_{A} A\cong A/Q,...
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2answers
32 views

Localization of a local ring

If $A$ is a local ring, $\mathfrak{m}$ is its maximal ideal, then is $A_{\mathfrak{m}}\cong A$ ? That's, in my opinion, because every denominator is invertible. Am I right?
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34 views

Vanishing set of a pullback section

Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$. If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and ...
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0answers
25 views

Is this localization of ideal correct?

If I have $$I=\left(x^2+y^2-yz,xyz-x,y(y-z)(yz-1)\right)\subset\mathbb{C}[x,y,z]$$ is then $I\mathbb{C}[x,y,z]_{(x,y)}$ equal to $\left(x^2+y^2-y,x,y\right)=\left(x,y\right)$? Thank you!
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0answers
60 views

How to visualize an ideal after localization?

Maybe I have not really understood what does it mean localization of a ring or a module at a prime ideal. I know the definition but i cannot really use it in the practice. If I have an ideal, in my ...
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1answer
48 views

Is it correct this way to compute that radical ideal?

Is it correct to compute that radical ideal in this way? $$\sqrt{(x^2,xz^2-x,y-z)}=\sqrt{(x^2,xz^2-x,y-z,x)}=\sqrt{(y-z,x)}=(x,y-z)$$ In particular, I added $x$ to generators inside the 'root' ...
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1answer
51 views

Radical of an ideal.

How can I compute $\sqrt{(x^2+y^2-1,yz-1)}$ as ideal of $\mathbb{C}[x,y,z]$? Actually I have to prove that $(x^2+y^2-1,yz-1)$ is prime but I don't know how. Could you give me some suggestions, ...
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1answer
52 views

flat f.g. modules over a commutative, local, Noetherian ring are free

I'm trying to prove that flat f.g. modules over a commutative, local, Noetherian ring are free. I think I've got really close to a proof, but I'm stuck at the last step that finishes the proof. So, ...
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0answers
38 views

Does the “Leibniz multicategory over $R$” have an accepted name?

Let $R$ denote a commutative ring. Definition. The "Leibniz multicategory" over $R$ is given as follows: Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate'). ...
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0answers
42 views

Does a commutative semilocal ring have only finitely many idempotents? [closed]

Does a commutative ring R with only finitely many maximal ideals, also has only finitely many idempotents? Also does R have to be finite direct product of indecomposable rings? Any pointers or ...
1
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1answer
39 views

$\mathrm{Frac}(R)/R$ as a direct limit

Let $R$ be an integral domain, let $F=\mathrm{Frac}(R)$ be its field of fractions. Then $R$ is a submodule of $F$, hence we have quotient module $F/R$. It is true that $$F/R \cong \varinjlim R/rR$$ ...
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0answers
47 views

Why is this intersection supported on the closed point?

Let $R$ be a (commutative unitary) local ring. Let $M$ and $N$ be finitely generated $R$-modules such that $\mathrm{length}(M\otimes_R N)$ is finite. Let $x$ be the closed point of $X=\operatorname{...
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1answer
35 views

A good reference for irreducible and noetherian spaces

I am looking for a comperhensive reference for irreducible and noetherian topological spaces. Also, a reference for prime spectrum of a commutative ring.
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1answer
24 views

Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By ...
2
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2answers
76 views

Quotient of a polynomial ring localized

Question: Prove that $\mathbb{R}[x,y]/(xy)$ localised at $(x-a)$ is isomorphic to the ring $\mathbb{R}[x]$ localised at $(x-a)$. Related question: What is the local ring at the point $(0,0)...
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1answer
40 views

Are formal power series rings over Dedekind domains formally smooth?

Let $A$ be a Dedekind domain. Consider the ring of formal power series $A[[t]]$ over $A$. Now let $B$ be any $A$-algebra, and let $N\subset B$ be a nilpotent ideal. Then, can any homomorphism $$A[[t]...