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

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9
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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 ...
3
votes
1answer
80 views

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? ...
4
votes
1answer
139 views

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,...
4
votes
2answers
248 views

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 ...
2
votes
2answers
146 views

Confusion about unique isomorphism $M \otimes N \to N \otimes M$

This is a follow up question to my previous question here. I'm confused about the following: in Atiyah-Macdonald they state that there exists a unique isomorphism $M \otimes N \to N \otimes M$, $m \...
5
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1answer
396 views

There exists a unique isomorphism $M \otimes N \to N \otimes M$

I want to show that there is a unique isomorphism $M \otimes N \to N \otimes M$ such that $x\otimes y\mapsto y\otimes x$. (Prop. 2.14, i), Atiyah-Macdonald) My proof idea is to take a bilinear $f: M \...
8
votes
1answer
915 views

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 ...
2
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1answer
381 views

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, $K$ ...
3
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2answers
218 views

Is this computation of the tensor product correct?

I'm reading the proof of the existence of the tensor product. If $M,N$ are two $R$-modules then we can construct the tensor product $T$ as the quotient $C/D$ where $C$ is the free module over $M \...
7
votes
2answers
162 views

An extension with the induced map on Spec being bijective.

Let $A$ be a commutative ring with unit. Let $A\subset A[b]$ be an extension of rings such that $b^n, b^m\in A$, where $m,n$ are positive integers that are coprime with each other. Show that $SpecA[b]\...
5
votes
2answers
218 views

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 ...
1
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1answer
101 views

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?
3
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1answer
115 views

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 ...
1
vote
1answer
77 views

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?
3
votes
1answer
188 views

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 ...
5
votes
2answers
234 views

What's so “shrieky” about this shriek map?

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 ...
1
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2answers
155 views

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 ...
1
vote
1answer
138 views

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 ...
1
vote
1answer
282 views

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 ...
6
votes
3answers
243 views

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 ...
12
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3answers
883 views

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 ...
6
votes
2answers
622 views

Krull dimension of local Noetherian ring (2)

Let $(A,\mathfrak{m})$ be a local Noetherian ring and $x \in \mathfrak{m}$. Prove that $\dim(A/xA) \geq \dim(A)-1$, with equality if $x$ is $A$-regular (i.e. multiplication with $x,$ as a map $A\...
4
votes
1answer
116 views

“standard” form of a finite morphism

Every etale morphism is locally (passing to affine neighbourhoods and then to their coordinate rings) of the form $A \to (A[x]/(P(x)))_b$ where $P(x)$ has the property that $P'(x)$ is invertible in $(...
1
vote
2answers
104 views

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 $A=K$.
0
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1answer
251 views

Rabinowitz trick and saturated ideals

Let $k$ be a field and $I\trianglelefteq k[X_1,\dots,X_n]=S$ an ideal, generated by $\{f_1,\dots,f_s\}$. Fix $f \in S$ and let $Y$ be a new indeterminate. Let $\tilde{I}=(f_1,\dots,f_s,1-fY)\...
6
votes
1answer
2k views

Saturated ideal

Let $k$ be a field, let $I \triangleleft k[X_1,\dots,X_n]=S$ be an ideal and fix $f \in S$. The saturated ideal of $I$ is $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \ s.t. \ f^mg \in ...
4
votes
1answer
825 views

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 ...
3
votes
1answer
1k views

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 ...
7
votes
2answers
1k views

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). ...
3
votes
2answers
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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.
0
votes
1answer
283 views

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 ...
6
votes
1answer
470 views

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 ...
6
votes
1answer
229 views

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 ...
11
votes
2answers
706 views

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 ...
9
votes
2answers
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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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votes
5answers
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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 ...
2
votes
2answers
1k views

Wanting to show $a+x$ is a unit for unit $a$ and nilpotent $x$ [duplicate]

Possible Duplicate: Units and Nilpotents If $a$ is a unit and $x$ is nilpotent, I'm trying to show that $a+x$ is a unit. Pf.: If $a$ is a unit, there exists a non-zero invertible element $a^{-...
0
votes
2answers
183 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{...
13
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1answer
560 views

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 ...
13
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4answers
330 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} \...
4
votes
1answer
110 views

If $g$ and $g\circ f$ are graded homomorphisms, must $f$ be graded?

Question Let $A$ be a graded ring (always commutative with identity) and $M,N$ and $P$ be graded $A$-modules. Let $f:M \longrightarrow N$ and $g:N \longrightarrow P$ be $A$-module homomorphisms ...
7
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1answer
200 views

The product of two spectral spaces

Notice: the following statements about the product topologies are all Cartesian product topology, we are in the category of topology not the category of schemes. In this page of sober space, it said ...
5
votes
1answer
361 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak m_i}M/\...
6
votes
2answers
274 views

How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$

Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{...
6
votes
3answers
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Does $A$ a UFD imply that $A[T]$ is also a UFD?

I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD. The only thing I am sure I could try to use is Gauss's lemma. Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\...
7
votes
3answers
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When the localization of a ring is a field

Let $R$ be a commutative noetherian ring with no nonzero nilpotents. Let $p$ be a minimal prime of $R$. Could you help me to prove that $R_p$ is a field?
2
votes
1answer
125 views

An equivalent condition for having finite length

Let $R$ be a commutative noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite length if and only if $M_p=0$ for every non-maximal prime ideals $p$?
8
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2answers
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Relation between projective modules over $R$ and $R[T]$

Let $R$ be a commutative ring and $R[U]$ the polynomial ring in one variable. What is the relation between projective modules over $R$ and projective modules over $R[U]$? Is every projective module ...
2
votes
3answers
85 views

If $Ra$ is free for $a\neq 0,$ is $a$ regular?

Let $R$ be a commutative ring with unity, and $0\neq a\in R.$ We will say that an element $x\in R$ is linearly independent if $\{x\}$ is a linearly independent set. A non-zero element of $R$ is called ...
3
votes
2answers
857 views

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 ...