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

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2
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2answers
85 views

Can a ring of integers contain a $2$-dimensional noetherian normal integral domain?

Let $K$ be a number field with ring of integers $O_K$. Does there exist a $2$-dimensional subring $A\subset O_K$? Clearly, if such a subring $A\subset O_K$ exists, we have that $A$ is an integral ...
2
votes
1answer
99 views

Irreducible polynomial over Dedekind domain remains irreducible in field of fractions

Let $\mathcal{O}$ be a Dedekind domain, $K$ its field of fractions. Suppose $f\in \mathcal{O}[X]$ is irreducible. Is it irreducible in $K[X]$? The motivation for my question is that this is true for ...
2
votes
3answers
450 views

Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$

Let $R$ be a noetherian ring and $M$ a finitely generated $R$-module with projective dimension $n$. Then for every finitely generated $R$-module $N$ we have $\mathrm{Ext}^n(M,N)\neq 0$. Why? By ...
2
votes
1answer
191 views

Formal power series ring over a valuation ring of dimension $\geq 2$ is not integrally closed.

I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory, but got stuck. The question is: If $R$ is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring ...
2
votes
2answers
655 views

What is a lift?

What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit ...
2
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1answer
178 views

Existence of an element of given orders at finitely many prime ideals of a Dedekind domain

Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = ...
2
votes
2answers
422 views

What is Hilbert polynomial of this projective variety?

Suppose you have a map $\varphi\colon\mathbb{C}^m\times\mathbb{C}^n\to\mathrm{Mat}_{m,n}(\mathbb{C})$ defined by sending $(\mathbf{u},\mathbf{v})\mapsto\mathbf{u}\cdot\mathbf{v}^T=(u_i,v_j)$. So ...
2
votes
0answers
166 views

Is an irreducible element still irreducible under localization?

Suppose $R$ is a domain. We say an element $x\in R$ is "irreducible" if $x=yz$ implies that $y$ or $z$ is a unit or both are units. I want to know if an irreducible element is still an irreducible ...
2
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3answers
719 views

How to determine whether a unique factorization domain is a principal ideal domain?

Could someone please provide an example of a unique factorization domain that is not a principal ideal domain? Furthermore, is there some way to determine whether a UFD is a PID?
2
votes
1answer
241 views

Help on a proof of a Theorem of Rees

I'm studying on this book http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false on page 10 there is a Rees Theorem. I'd like to know why the ...
2
votes
3answers
298 views

Is a regular sequence ordered?

A regular sequence is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring $$ \frac R ...
1
vote
2answers
96 views

Extending an automorphism to the integral closure

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of ...
1
vote
2answers
63 views

An exercise on tensor product over an integral domain.

This post is the natural conclusion of another one (An exercise on tensor product over a local integral domain.). Let $M$ be a finite module over an integral domain $A$. Let $Q$ be its fraction ...
1
vote
1answer
113 views

non-examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of ...
1
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2answers
220 views

Ideals-algebraic set

Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3) = \{ (t,t^2,t^3) \mid t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3)) = \langle Y-X^2,Z-X^3 \rangle$$ Finally, prove that the ...
1
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0answers
133 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
1
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1answer
120 views

$\Gamma_{\mathfrak a}(I)$ is an injective $R$-module for every injective $R$-module $I$

Is there a proof for Proposition 2.1.4 of Local Cohomology book by Brodmann-Sharp not using Artin–Rees Lemma? Proposition 2.1.4: Let $I$ be an injective $R$-module. Then $\Gamma_{\mathfrak a}(I)$ ...
1
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1answer
314 views

Isomorphism from $B[y]/IB[y]$ onto $(B/I)[y]$

For some reason I can't crack the following problem: Let $B$ be a ring, $I$ an ideal, and $A := B[y]$ the polynomial ring. Construct an isomorphism from $A/IA$ onto $(B/I)[y]$. How to ...
1
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2answers
91 views

intersection of non zero prime ideals of polynomial ring R[x] over integral domain R is zero

Let R be an integral domain. Then how to show that intersection of non zero prime ideals of R[x] is zero.
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2answers
72 views

$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$

I'm doing this exercise. Let $A$ be an integral domain, then prove that $$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} ...
1
vote
1answer
176 views

Support of $\operatorname{Hom}(R/I, M)$

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $M$ be an $R$-module. Is the following formula true? $\operatorname{Supp}\operatorname{Hom}(R/I, M)=\operatorname{Supp}(M) \cap V(I)$ If ...
1
vote
1answer
153 views

Height unmixed homogeneous ideal and a non-zero divisor

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an ...
1
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2answers
41 views

Can we contruct a basis in a finitely generated module

Let $M=\langle e_1,\ldots,e_n\rangle$ be a finitely generated $R$-module. My question is can we construct a free submodule $F$, i.e, isomorphic to $R^s$ for some $s$, finding a subset ...
1
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1answer
314 views

Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, ...
1
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2answers
315 views

Prime ideals in tensor products of algebras and their pullbacks

Suppose $\mathfrak{p}$ is a prime ideal in $B\otimes_CA$, and $\mathfrak{p}_A,\mathfrak{p}_B,\mathfrak{p}_C$ are its pullbacks in $A,B,C$. Does it hold: $(B\otimes_CA)_{\mathfrak{p}}\cong ...
1
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2answers
103 views

isomorphic ideals and projective dimensions of quotients

Let $R$ be a Noetherian ring and $I,J$ proper ideals that are isomorphic as $R$-modules. Can we conclude that the projective dimensions of $R/I,R/J$ are equal?
1
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1answer
172 views

Help with these isomorphisms

Let $X$ be an affine algebraic set and $f\in K[X]$ where $K[X]$ is the coordinate ring of $X$. Suppose $I(X)=\langle G_1,\ldots,G_r\rangle$ and $W=Z(G_1,\ldots,G_r,FT_{n+1}-1)$, where ...
1
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1answer
133 views

Krull dimension of this local ring

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
1
vote
1answer
220 views

existence of finite free resolutions of finite modules over polynomial rings

Theorem 2.8 in Chapter XXI of Lang's Algebra says Theorem 2.8. Let $R$ be a commutative Noetherian ring. Let $x$ be a variable. If every finite $R$-module has a finite free resolution, then every ...
1
vote
1answer
68 views

Normal at every localization implies normal

I'm having some trouble with basic ring theory. Let $A$ be an integral domain and $\alpha$ an element of its fraction field integral over $A$. I am trying to understand a proof that $\alpha\in A$ ...
1
vote
1answer
285 views

Dimension of a tensor product of affine rings

The dimension of a ring is defined as the length of a longest prime chain as usual. Let $A,B$ be affine rings over a field $k$. Then $$\dim A\otimes_k B = \dim A + \dim B.$$ How can we prove or ...
1
vote
1answer
178 views

comaximality of ideals in a commutative ring with unit

Suppose we have a commutative ring $R$ with unit. I'm curious about what condition(s) on $R$ would be sufficient (without Axiom of Choice) to give a converse to the following familiar result: (#) If ...
1
vote
1answer
121 views

Surjective homomorphism in Laurent polynomial ring.

Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Consider $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$ where $a_i\in \mathbb C\setminus \{0\}$ and let $I$ be the ideal ...
0
votes
0answers
48 views

Is an irreducible ideal in $R$ irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
0
votes
0answers
39 views

intuitive interpretation of the multiplicity

Although logically I can understand and use multiplicity (for defi􀀀nition see 4.1.5 of Bruns_Herzog), yet, the concept of multiplicity of a module is not completely clear for me. Is there an ...
0
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0answers
133 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? if 1) $R$ is non-Noetherian ring , 2) $R$ is Noetherian ring and $M$ ...
0
votes
2answers
264 views

Nilradical and Jacobson's radical. [duplicate]

Let A be a commutative ring with 1. 1) Prove that a sum of a nilpotent element and an invertible element is invertible. 2) Prove that if $f=a_0+a_1x+\dots+a_nx^n \in A[x]$ a) $\exists f^{-1}\in ...
0
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3answers
1k views

Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain.

Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain. We haven't really discussed principal ideal domains. I've heard that this is easy, but I just ...
11
votes
1answer
458 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
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3answers
544 views

Galois ring extension

Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.
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0answers
363 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
9
votes
2answers
632 views

Rings whose spectrum is Hausdorff

Let $A$ be a commutative ring with $1$ and consider the Zariski topology on $\operatorname{Spec}(A)$. When will $\operatorname{Spec}(A)$ be a Hausdorff space? If $A$ has positive or infinite ...
8
votes
3answers
486 views

A fraction field is not finitely generated over its subdomain

I'm looking for proofs of the following fact. Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as $R$-module. I know this fact is ...
8
votes
1answer
224 views

Why is UFD a Krull domain?

Matsumura mentions this as if it is obvious, and I can't find this result anywhere. Am I missing something obvious here?
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1answer
824 views

Recalling result of tensor product of polynomial rings

Let $k$ be a field (alg closed if you want). Now let $I_{i}$ be an ideal of $k[x_{i}]$ for every $i \in \{1,2,\ldots,n\}$. Is it always true that: $$k[x_1,x_2,\ldots,x_n]/ \langle I_1,I_2,\ldots,I_n ...
6
votes
1answer
295 views

question about typical proof of Krull Intersection Theorem

In Atiyah-Macdonald, in the proof of Theorem 10.17 (Krull's intersection theorem), the authors go through a 4-line chain of arguments to show that that the kernel $E=\bigcap_{n=1}^{\infty}\mathfrak ...
6
votes
1answer
140 views

Verifying Hilberts Nullstellensatz on a particular example

Let $k$ be an algebraically closed field of characteristic $2$ and consider the following equations: $$xy + z^2 = 0$$ $$uv + w^2 = 0$$ $$uy + vx = 0$$ It's not hard to parameterize solutions to these ...
6
votes
0answers
189 views

Punctured plane algebraic over a finite field?

I've been asked to prove, by my algebraic geometry teacher, that the punctured affine plane $\mathbb{A}_k^2 \backslash \{0,0\}$ is not an algebraic set, i.e. is not the zero set of any set of ...
6
votes
1answer
346 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
5
votes
0answers
138 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...