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

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Zariski Topology on Primary Spectrum $P$-$\operatorname{Spec}(R)$

So there's a proposition in Hummadi's journal Primary Spectrum I'd like to ask. It is said if given $R$ principal ideal domain and $a,b \in R$, then $D_p(a) \cap D_p(b) \supseteq D_p(ab)$, and $D_p(...
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1answer
73 views

If $l(B,A)$ is a prime ideal then $B$ is maximal in $A$

Let $B\subset A$ be commutative rings with identity. Furthermore $B$ is a domain. We are given the set $$l(B,A) = \{ b\in B\setminus \{0\}: B[b^{-1}] = A[b^{-1}]\} \cup \{0\},$$ where $B[b^{-1}]$ ...
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1answer
106 views

The relation between minimal ideals and zero divisors [duplicate]

How we can prove this Theorem. Let $R$ be a reduced ring. Then $a\in ZD(R)$ (the zero divisors of $R$) if and only if $a\in P$ for some minimal prime ideal $P$.
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1answer
45 views

Why is this polynomial a function of $X^3$?

In studying that recent question, I noticed that curious (or perhaps not so curious) property : if $x,y$ are rational numbers and $a$ is the real part of a cubic root of $x+iy$, then $Q(a^3)=0$ where $...
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74 views

If $x_i$ generate an $A$-module $M$, why do $1 \otimes x_i$ generate the extension of scalars $B \otimes_A M$?

In the following, let "ring" be a synonym for "commutative ring with identity". For rings $A, B$ and an $A$-module $M$, let $M_B = B \otimes_A M$ be the $B$-module obtained from $M$ by extension of ...
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2answers
66 views

How to prove $x$ doesn't lie in $R_M$

Let $R$ be an integral domain. $K$ is the field of fractions of $R$. Let $x=a/b \in K-R$ and $a \notin (b)$. How do I prove $x \notin R_M$ where $M$ is a maximal ideal containing $b$? The statement is ...
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127 views

Ring of fractions problem

How do I can determine the ring of fractions of $\mathbb{Z}[X]$ ? I don't know the process that I have to follow for do it.
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69 views

Kernel of canonical morphism in inductive limit (proof by induction)

Let $\langle I, \leqslant \rangle$ be a directed poset and $\langle M_i, \mu_{i,j} \rangle$ be a directed system of $A$-modules over $I$. Now let $$ C = \bigoplus\limits_{i \in I} M_{i}, $$ and $D$ ...
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1answer
100 views

Monomial ordering problem

I've got the following problem: Let $\gamma$, $\delta$ $\in$ $\mathbb R_{> 0}$. The binary relation $\preceq$ on monomials in $X,Y$ is defined: $X^{m}Y^{n} \preceq X^{p}Y^{q}$ if and only if $\...
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61 views

The point $(0,1)$ on coordinate axis, and any point on a single line look the same. [whereas $(0,0)$ is not]

I am trying to show that $S_{1}^{-1}[K[X,Y]/(XY)]$ is isomorphic as rings to $S_{2}^{-1}K[X]$, where $K$ is a field, $S_{1}=(k[X,Y]/(XY))\setminus(\bar{X},\bar{Y}-1)$, and $S_{2}={\{1,X,X^2,\dots}\}$....
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1answer
52 views

Two different definitions of Derivation

Here are two definition of derivations Definition 1 Let $A \rightarrow B$ be a homomorphism of commutative algebras, and $M$ a $B$-module. We define the derivations $...
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1answer
208 views

Hypotheses of the Conormal Exact Sequence

On Wikipedia, in the description of the conormal exact sequence, it is described as arising from a closed immersion, which corresponds in the affine case to a surjection of algebras. However, in ...
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1answer
45 views

If ideals $Q_1,Q_2$ lie over a prime in $\Bbb{Z}$ their product lies over the prime squared?

Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ...
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1answer
196 views

Problem on two ideals and their quotient

For two ideals $I$ and $J$ in a commutative ring $R$, define $I : J = \{a\in R : aJ \subset I\}$. In the ring $\mathbb{Z}$ of all integers, if $I = 12\mathbb {Z}$ and $J = 8\mathbb {Z}$, find $I : J$. ...
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1answer
81 views

On the dimension of a module over a Noetherian ring

For a noetherian ring $R$, let $x\in R$ be a non-zerodivisor of an $R$-module $M$. Then this is a well-known fact: $$\dim M/xM \leq \dim M-1.$$ I saw a proof for the case of finitely generated $M$. ...
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1answer
119 views

$u\in R$ is a unit iff $u+x$ is a unit for all $x\in \mathcal{N}(R)$

Let $R$ be a commutative ring with identity and denote by $\mathcal N(R)$ its nilradical. It is known that an element $u\in R$ is a unit if and only if $u+x$ is a unit for all $x\in\mathcal N(R)$. In ...
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1answer
252 views

Semicontinuity of fiber dimension

The following two on commutative algebra are true? Let $S$ be a f.g. algebra over a field $k$. Let $e$ be an integer. Then (1) There is an ideal $I\subset S$ such that if $Q$ is a maximal ideal of $S$...
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2answers
182 views

Radical and nilrad

I'm trying to prove that the set $\mathrm{nilrad}(A)$ of nilpotent elements of $A$ is an ideal Pf/ if $g\in\mathrm{nilrad}(A)$, then $g^n = 0$, for some $n>0$. Let $h$ be an element of $\mathrm{...
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3answers
284 views

Example where $\operatorname{grade}(I,M)>\operatorname{height} I$

Let $I$ be an ideal of a noetherian ring $R$ and let $M$ be a finite $R$-module. We need to show if $I$ is generated by $n$ elements, then $\operatorname{grade}(I,M)\le n$. Could any one give an ...
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1answer
183 views

homomorphism of Laurent polynomial ring

This question is similar to the question link. Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Given $r\in \mathbb Z_+$ and $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$, ...
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1answer
52 views

If $Rm$ is free and $N$ is free is $ m \otimes n \neq 0$?

This post is a follow up to the counterexample presented in the following questions If $Rm$ is free, how do you show $m \otimes n \neq 0$?. The hope now is that we can eliminate the pathological ...
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1answer
173 views

faithful, finitely generated module over a local ring

Let $A$ be a commutative local ring, with unique maximal ideal $\mathfrak{m}$, and residue field $k:=A/\mathfrak{m}$. Let $M$ be a faithful, finitely generated $A$-module. If $M/\mathfrak{m}M$ is 2-...
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1answer
49 views

Roots of Units in Complete $\mathbb{C}$-Algebras

$\newcommand{\cc}{\mathbb C}$ Let $R$ be a finitely generated $\cc$-algebra and ${\frak m}\subset R$ a maximal ideal. Denote by $\hat R$ the completion of $R$ with respect to $\frak m$. Assume that $x\...
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3answers
316 views

rank function on Spec (help with definition)

one definition of the line bundle over a ring is: a finitely generated projective A-module such that the rank function Spec A → N (positive integers) is constant with value 1. We call A itself the ...
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1answer
51 views

Applications of the Formal Laurent Lattice

Attach a (Laurent) monomial weight $x_1^{i_1} \cdots x_n^{i_n}$ to each point $(i_1, \dots, i_n)$ of $\mathbb{Z}^{n}$ and call it $\mathbb{Z}^{n}[x_1, x_{1}^{-1}, \dots, x_n, x_n^{-1}]$. Does this ...
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0answers
21 views

What properties are preserved by direct limits? [on hold]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
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0answers
14 views

Prove that integral closure of $\mathbb R[x,y]/(y^2-x^3-x^2)$ is $\left( \mathbb R[x,y]/(y^2-x^3-x^2) \right) \left[ \frac{y}{x} \right]$ [duplicate]

i have to give a proof of the Headline. I just showed, that $y/x$ is integral over $R:=\mathbb R[x,y]/(y^2-x^3-x^2)$. How do I show, that $\bar R = R[t]$ where $t=y/x$? Furthermore, I have to show, ...
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1answer
48 views

Is it true that $\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x)$? [on hold]

I need to show that $(xy^2-1)$ is prime in $\mathbb{Q}[x,y]$ and I tried to consider that isomorphism. Does it hold? Thank you.
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0answers
34 views

Prime spectrum of tensor product of two R-algebras [on hold]

Let $R$ be a commutative ring and $A_1$ and $A_2$ two commutative unital $R$-algebras. Is there any characterization for $\mathrm{Spec}(A_1\otimes_R A_2)$? Or how can we deduce that $ \mathrm{Spec}(...
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1answer
48 views

Finite type + integral = finite

Let $A \subseteq B$ be rings (comm. with unity). I am struggling to see why the following equivalence holds for $B$ interpreted as a $A$-Algebra: $A \rightarrow B$ is of finite type and $A\...
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0answers
31 views

Structure constants in a finitely generated $\mathbb{k}$-algebra

Let $\mathbb{k}$ be a field of characteristic $0$. Suppose we have a finitely generated graded $\mathbb{k}$-algebra $A= \bigoplus_{i=0}^{\infty}A_i$ which is free of finite rank as a module over a ...
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0answers
28 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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0answers
56 views

Dimension of polynomial rings and tensor products of residue fields

In Matsumura textbook to show that $\dim A[x] = \dim A + 1$, first it states that $A[x] \otimes k(\mathfrak{p}) = k(\mathfrak{p})[x]$ which is one dimensional. Then it uses the theorem 15.1.(ii) since ...
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0answers
52 views

primary decomposition of injective envelope of a module

The Exercise A3.6 of Eisenbud's book, Commutative Algebra with a view Toward Algebraic Geometry, is: Assuming that $R$ is Noetherian, let $M$ be any finitely generated $R$-module. a. Let ...
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1answer
30 views

How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$?

How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$? I know the answer in the sense that i know that $(2x+1)$ is a maximal principal ideal of that polynomial ring. But if i ...
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0answers
26 views

Singular ideal of an idealization

Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+...
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0answers
41 views

Motivation for localization as given in Eisenbud

Eisenbud writes that the affine ring $A(X-Y)$ is obtained from $A(X)$ by adjoining a multiplicative inverse of $f$, where $Y$ is the vanishing set of the function $f$. $A(X-Y)$ is the set of ...
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34 views

grading of the tensor product

I have just had a look at http://therisingsea.org/notes/GradedModules.pdf to look up the grading of the tensor product of two graded modules over a graded ring (see page 10). And I am wondering, why ...
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48 views

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 ...
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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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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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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 ...
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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
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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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 $\mathbb{Z}/p^k\mathbb{Z}$ ...
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0answers
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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0answers
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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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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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'). ...