# Tagged Questions

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

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### A concrete example of an ideal $I\subseteq K[x_1,\ldots,x_n]$ and coprime polynomials $f,g$ such that $(I,f)\cap (I,g)\neq (I,fg)$

I know that in a polyomial ring $K[x_1,\ldots,x_n]$ over a field $K$, given a monomial ideal $I$ and two coprime monomials $f,g\notin I$, it holds $$(I,f)\cap (I,g)=(I,fg)$$ However, I've been ...
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### If $X(F) \cap Y$ is dense in $Y$, then $Y$ is defined over $F$.

Let $k$ be an algebraically closed field, $F$ a subfield of $k$, $A$ a finitely generated, reduced $k$-algebra, and $A_0$ an $F$-subalgebra of $A$, of finite type over $F$, such that the canonical $k$-...
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### Homological dimension of categories of modules

Let $A$ be a Noetherian ring. We have two categories: (a) category of $A$-modules (b) category of finite type $A$-modules. Do their homological dimensions agree? The homological dimension of an ...
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### When $(f(T),f'(T))=R[T]$?

Let $R$ be a UFD, $f(T) \in R[T]$ a monic polynomial of degree $d \geq 2$, and $f'(T)$ the formal derivative of $f(T)$. When the ideal generated by $f$ and $f'$ equals $R[T]$? (If $d=1$, then $f'=1$,...
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### Smooth algebras

All rings are Noetherian and eft. An $A$-algebra $B$ is smooth if it is flat and the fibres are geometrically regular. I want to see some examples of this notion. So I considered the $\mathbb Z$-...
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### Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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### Form of the elements of a localization

If I have a ring $R$, a multiplicatively closed subset $U\subset R$, and consider an element of the localization: $\frac{r}{r'} \in U^{-1}R$, can I then assume without loss of generality that $r'\in U$...
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### Is every unramified extension of DVRs simple?

Let $A$ be a discrete valuation ring with maximal ideal $\mathfrak{m}$, fraction field $K$, and $L$ a finite separable extension of $K$ degree $n$, unramified w.r.t. $A$. Let $B$ be the integral ...
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### How can I compute completions of rings?

I want to learn about how to compute the completions of local rings. For example, I want to be able to compute the completions of \begin{align*} \left(\frac{\mathbb{C}[x,y]}{(y^2 - x)}\right)_{(x,y)} ...
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### Isomorphism of modules [duplicate]

Are $\mathbb{C}[x,y]/(x,y)$ and $\mathbb{C}[x,y]/(x-1,y-1)$ isomorphic as $\mathbb{C}[x,y]$-modules? I think they are cyclic so they are isomorphic, but I'm not sure.
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### Faithfully flat descent of projectivity and freeness

I am reading this paper. It is proven there that if $f:A\rightarrow B$ is a faithfully flat morphism of rings and $M$ an $A$-module such that the $B$-module $M\otimes_A B$ is projective, then $M$ ...
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### Homogeneous System of Parameters

Assume that $R$ is a finitely generated graded $k$-algebra of Krull dimension $n$. Is it true that any set $\{f_{1},f_{2},...,f_{n}\}$ of homogeneous algebraically independent polynomials is a ...
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### A condition that the ratio of locations is maximal

Sea $R$ un anillo conmutativo con identidad e $I$ un ideal de $R$ y $m$ un ideal maximal de $R$. Mostrar que $\displaystyle\frac{R_m}{I_m}\neq{0}$ si y solo si $I\subseteq{m}$. Dm: $[\Rightarrow{}]$. ...
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### Combinatorial commutative algebra

Let G is simple graph and Δ(G) is clique complex, IΔ(G) has a 2-linear resolution if and only if, for any subset W ⊂ [n], one has H˜i(Δ(G)W ; K) = 0 unless i=0 (from comment): I do not understand ...
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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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### Example of non-noetherian ring whose spectrum is noetherian and infinite

A topological space is noetherian if it satisfies the descending chain condition for its closed subsets. Let be $R$ a commutative ring and let $\mathrm{Spec}(R)$ its spectrum with Zariski topology. I ...
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### In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
I am searching for a prime ideal of the ring $R=∏_{n=2}^{∞} {\mathbb Z}_{2^n}$ which is not maximal. In fact, since each ${\mathbb Z}_{2^n}$ is local with $\left<\bar 2\right>$ as the maximal ...
### Injectivity of $R \to R[t]/(f)$ for non-constant $f\in R[t]$
Question: Let $R$ be a (unital commutative) ring and $f = a_0 t^n + \cdots + a_n \in R[t]$ a non-constant polynomial. What are (necessary and sufficient) conditions on the coefficients \$a_0,\ldots,a_n ...