# Tagged Questions

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

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### Prime ideals in the ring of algebraic integers

Let $\mathcal{O}$ be the ring of all algebraic integers: elements of $\mathbb{C}$ which occur as zeros of monic polynomials with coefficients in $\mathbb{Z}$. It is known that $\mathcal{O}$ is a ...
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### Is this graded ring problem corrected?

let $B=\oplus_{i\geq0}B_i$ be a graded ring with $B_d=B_1^d$ for every $d\geq1$. Suppose $B_1$ is a finitely generated $A$-module for some ring $A$. Then, is $B$ an $A$-algebra in some canonical way? ...
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### Is “algebraic-variety” a relative concept?

Let A, B be two NON-isomorphic finitely generated k-algebras, is it possible they isomorphic as abstract commutative unitary rings? (any concrete examples?) If the answer to the above is possibly yes,...
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### Going down theorem with modification.

Going-down Thm: Let $A\subseteq B$ be an integral extension. Assume that $B$ is an integral domain and that $A$ is integrally closed. Then going down holds for the above extension. Question1: Can we ...
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### Existence of valuation rings in an algebraic function field of one variable

The following theorem is a slightly modified version of Theorem 1, p.6 of Chevalley's Introduction to the theory of algebraic functions of one variable. He proved it using Zorn's lemma. However, Weil ...
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### Is the integral closure of a local domain in a finite extension of its field of fractions semi-local?

If the answer is negative, I wonder under what conditions it would be semi-local. EDIT Here's an example of a local domain which is not necessarily a Japanese ring. Let $A$ be a valuation ring, $K$ ...
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### If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's On ...
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### Integrally Closed implies reduced

Suppose $A$ is a commutative ring with unit and that $A$ is integrally closed in $A[x]$. Show that $A$ is reduced?
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### Fields as a reflective subcategory of integral domains?

A subcategory $\mathbf A$ is reflective subcategory of $\mathbf B$ if for every $B\in\mathbf B$ there exists an $A_B\in\mathbf A$ and a $\mathbf B$-morphism $r_B \colon B \to A_B$ such that: for any ...
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### For a reduced ring $A$, must $A[x]\setminus A$ be multiplicatively closed?

Suppose $A$ is a reduced commutative ring. Is $A[x]\setminus A$ is multiplicatively closed?
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### Localizations of an integral domain with respect to finite prime ideals

I think the following proposition is likely to be true. I'd like to know a proof of it if any. Proposition Let $A$ be an integral domain, $K$ its field of fractions. Let $P_1, ..., P_n$ be prime ...
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On page 88 of Atiyah-Macdonald's "Introduction to Commutative Algebra" there is an exercise about the Grothendieck group $K(A)$ of a noetherian ring $A$. In this context to every finite ring ...
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### Polynomial algebra

Let $k$ be a field and $k \subset A \subseteq k[X]$ be a $k$-subalgebra of $k[X]$. Prove that $\dim(A)=1$ (Krull dim) and that $A$ is a finitely-generated $k$-algebra. My initial thought: Consider ...
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### Subring of polynomials

Let $k$ be a field and $A=k[X^3,X^5] \subseteq k[X]$. Prove that: a. $A$ is a Noetherian domain. b. $A$ is not integrally closed. c. $dim(A)=?$ (the Krull dimension). I suppose that the first ...
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### A proposition on a Dedekind domain

I need a proof of the following proposition(?). Actually I think I came up with a proof. But it's nice to confirm it and/or to know other proofs. Thanks. Proposition Let $A$ be a Dedekind domain. Let ...
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### Axiom of Choice (for example in the Snake Lemma)

If we have to make a choice, but in the end it doesn't matter what choice we made, did we really make a choice to begin with? More explicitly, somewhere in the standard diagram-chasing proof of the ...
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### Ring of holomorphic functions

Am I correct or not? I think that a ring of holomorphic functions in one variable is not a UFD, because there are holomorphic functions with an infinite number of $0$'s, and hence it will have an ...
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### Quotient field of a domain

Let $A$ be a commutative domain and $K=Quot(A)$, its field of fractions (quotient field). Prove that $K$ is a f.g. $A$-module if and only if $A=K$.
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### Understanding the conductor ideal of a ring.

Consider the inclusion of a ring $A$ into its integral closure $B$. The conductor ideal $I$ is defined as $I:=\{a\in A~|~aB\subseteq A\}$. This is supposed to describe the locus where the ...
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### A surjective homomorphism between finite free modules of the same rank

I know a proof of the following theorem using determinants. For some reason, I'd like to know a proof without using them. Theorem Let $A$ be a commutative ring. Let $E$ and $F$ be finite free modules ...
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### Krull dimension of local Noetherian ring

Let $R$ be a commutative local Noetherian ring and $\mathfrak{m}$ its maximal ideal. Prove that, if $\mathfrak{m}$ is principal, then $\mathrm{dim}(R)\leq 1$ (the Krull dimension of the ring). ...
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### Jacobson radical of $R[X]$, where $R$ is domain

Let $R$ be a commutative domain. Prove that the Jacobson radical of $R[X]$, i.e. the intersection of all maximal ideals, is the zero ideal. Thank you.
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### Reduced ring is integrally closed in polynomial ring

Let $R$ be a commutative ring, with 1. Prove that if $R$ is reduced, then $R$ is integrally closed in $R[X]$, i.e. $R \subset R[X]$ is an integral extension of rings. I found this problem in many ...
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### Is every proper nontrivial ideal in a Noetherian ring not flat?

I guess my general question is exactly what's in the title, but let me explain why I'm asking and how I came to it. Consider the ideal $I=\langle x,y \rangle \subset k[x,y]$ for a field $k$. Just to ...
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### Points and sheaf of functions for some schemes

I am (slowly) reading Eisenbud and Harris and trying to get my head around (affine) schemes. Let $R$ be a ring, and $X=\text{spec}(R)$, as usual with the Zariski topology. We have a basis of open ...
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### Examples demonstrating that the finitely generated hypothesis in Nakayama's lemma is necessary

Recall that Nakayama's lemma states that Let $R$ be a commutative ring with unity, and let $J$ be the Jacobson radical of $R$ (the intersection of all the maximal ideals of $R$). For any finitely ...
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### Is fibre product of varieties irreducible (integral)?

Let $k$ be an algebraically closed field and $X,Y$ varieties (i.e. integral, separated schemes of finite type over $k$). Is the fibre product $X \times_k Y$ necessary irreducible or integral? I ...
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### How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
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### Exercise 2.17(d) of Eisenbud's Commutative Algebra

First some notation: Let $P$ be a homogeneous prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a ...