# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $u\in R$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $\mathbb{Z}/p^k\mathbb{Z}$ ...
### 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'). ...