# Tagged Questions

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

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### Principal prime ideals are minimal among prime ideals in a UFD

Fulton, "Algebraic Curves," Exercise 1.39(a): Let $R$ be a UFD, and $P = (t)$ a principal, proper, prime ideal. Show there is no prime ideal $Q$ with $0 \subset Q \subset P$. After being ...
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Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. We define $L$-radical of $I$, denoted by $\sqrt[L]{I}$, the intersection of all primary ideals of $R$ containing $I$. It is ...
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### UFDs are integrally closed

Let $A$ be a UFD, $K$ its field of fractions, and $f$ an element of $A[T]$ a monic polynomial. I'm trying to prove that if $f$ has a root $\alpha \in K$, then in fact $\alpha \in A$. I'm trying to ...
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### Alternate proof of a corollary about integral dependence of rings

I am working on an alternative proof of Corollary 5.9, p.61 in Atiyah - MacDonald, "Introduction to Commutative Algebra". The Corollary reads as follows: "If $A \subseteq B$ are rings, $B$ is ...
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### Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.

Let $R$ be a local ring, and let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be an $R$-module. I understand that $M \otimes_R (R / \mathfrak m)$ is isomorphic to $M / \mathfrak m M$, but I ...
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### Factoring a ring homomorphism

From Atiyah-Macdonald, bottom of page 9: "Let $f: A \to B$ be a ring homomorphism. ... We can factorize $f$ as follows: $$A \xrightarrow{p} f(A) \xrightarrow{j} B$$ where $p$ is surjective and $j$...
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### localization of rings and polynomial functions

Let $f$ and $g$ be two polynomials (polynomial functions in $n$ variables); if in some localization of the ring $k[X_1,\ldots, X_n]$ exists the class $\frac{f}{g}$, it defines in a unique way the ...
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### Weighted initial ideal versus lex or graded reverse lex initial ideal

By imposing certain weights $\mathbf{w}$ on the variables, say, of a polynomial ring $k[x_1,\ldots, x_n]$, I read that we may obtain the initial ideal $in_{\mathbf{w}}(I)$ of an ideal $I$ with respect ...
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### Finite extension of residue fields of DVR's

Let $R$ be a DVR with $K = Quot(R)$ and residue field $k$. Let $k'/k$ be a finite field extension. I would like to have a reference for the following statement (or to see, that it is not true): There ...
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### Tensor product of $\mathbb R$ and $\mathbb C$ over $\mathbb R$.

$$\mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} = \;?$$ I guess this guy is just $\mathbb{C}$, is this correct?
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### Computing the local ring of an affine variety

Let $W=V(y^{2}-x^{3}) \subseteq \mathbb{A}^{2}$ and $k$ algebraically closed. Clearly the dimension of the tangent space at the origin is $2$. I want to compute this using the definition the fact that ...
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### Fields finitely generated as $\mathbb Z$-algebras are finite?

Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra. (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite?
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### Radical of annihilator of a module

I met the following problem when I studied graded ring theory. I have no idea to solve it. Please help me. Thank you very much ! Let $R$ be a commutative $\mathbb{Z}$-graded ring, $M$ is a graded R-...
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### A yet another theorem on the different ideal of algebraic number fields

I think I came up with a proof of the following theorem using non-archimedian completions. But I'm not 100% sure. Is this correct? Theorem Let $A$ be a Dedekind domain, $K$ its field of fractions. ...
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### Coheight of an ideal generated by $n$ elements

Let $R$ be a noetherian ring, and let $I$ be a proper ideal in $R$. If $I$ is generated by $n$ elements, we have by Krull's Principal Ideal Theorem that the height of $I$ is at most $n$. Is it true ...
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### Is every Noetherian module finitely generated?

I was just wondering whether the following statement is correct. Let R be a ring and M a noetherian R module. Then M is finitely generated.
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### What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
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### If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$

$\lhd$ will stand for "is an ideal of" in this post. Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$ I don't think it does, but I'm having difficulty finding a ...
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### Detecting whether something is a Dedekind domain

Consider the three rings $\mathbb{C}[x,y] / \langle x^4 + xy -1\rangle$, $\mathbb{Z}[x,y] / \langle x^4 + xy -1\rangle$ and $\mathbb{F}_2[x,y] /\langle x^4- y^3 \rangle$. I am supposed to detect ...
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### How to show a ring is normal or not, and how to show the normalisation of the ring

I am confused about how to show whether a ring is normal or not. For example, consider the $k$-algebra $k[x,y] /\langle x^2 - y^3 \rangle$, which is a domain. How do I show it is not normal? Are ...
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### Computing an example of Ext

Let $k$ be a field. I want to compute $\operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k)$. However I have no idea how to do this? I cannot even think how to construct a projective resolution ...
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### Finitely generated projective module

Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free? What I know about a finitely generated projective module $M$ over a PID $R$ is ...
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### Question on Noetherian/Artinian properties of a graded ring

Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the ...
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### Ascending chain conditions on homogeneous ideals

Here is one exercise from some notes on graded rings. I tried but I got no idea to solve it. Please help me. Thanks. Let $R$ be a graded ring. Prove that $R$ is Noetherian (Artinian) if and only ...