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

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transcendence degree of rees ring

Let R be a ring wich is a domain and I an ideal of R .How can I compute the tr.deg of the rees ring (R(I)) over R? in this way I want to check the altitude formula.
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1answer
21 views

A question on smooth morphisms and 'fiberwise' smooth morphisms

Let $X$ be a scheme, $x\in X$ a point and $f\colon \operatorname{Spec}(k(x))\to X$ the canonical morphism. Is $f$ always a smooth morphism? Now suppose $g\colon X\to Y$ is a scheme over some ...
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1answer
21 views

Two points in a proof of regularity of $R/I$

In the proof of the fact that "if $I$ is an ideal of the regular local ring $(R,m)$ such that $R/I$ is regular then $I$ can be generated by part of a minimal generating set of of $m$", I saw in a ...
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15 views

Existence of maximal homogenous ideals strictly contained in the positive part

Let $A$ be a ($\mathbb{N}$-)graded ring. Set $A_+ = \bigoplus_{d \geq 1} A_d$. Proposition 13.2 in Gortz-Wedhorn says that the set $\mathcal{I} = \{ \text{homogenous ideals strictly contained in ...
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0answers
11 views

Do lattices in a field of fractions contain an ideal?

Let $R$ be a noetherian commutative integrally closed domain whose field of fractions $K$ is a finite extension of the field of fractions $Q$ of $\Lambda = \mathbb{Z}_p[[T]]$. Let $L \subset R$ be a ...
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51 views

Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
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1answer
58 views

$\mathbb{Q}[x,1/x]$ is normal?

Let $x$ be a transcendental. I heard $\mathbb{Q}[x,1/x]$ is a normal domain. But I don't understand why. Help me, thanks.
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1answer
132 views

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[x,y]/(x^2+y^2+1)$$ is ...
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1answer
27 views

Basis for the completion of a free module

This (or similar) question might have been asked before- apologies for any duplication. I've got a Dedekind domain $R$, a non-zero prime ideal $P$ of $R$ and the completion $\widehat{R}$ of $R$ wrt ...
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1answer
81 views

The Zariski topology on $\mathrm{spec} \ A$ as an intial topology

Given any ring $A$ let $\mathrm{spec} \ A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical maps from ...
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1answer
70 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
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39 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
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1answer
34 views

In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
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1answer
46 views

Does $\operatorname{Hom}(M,T)\cong\operatorname{Hom}(N, T)$ for all $A$-modules $T$ mean $M\cong N$?

The question is contained in title, I'm working with $A$-modules $M$ and $N$. I feel like Yoneda's lemma is what I'm looking for but it applies to functors into the category of sets, whereas ...
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55 views

Atiyah & Macdonald's Introduction to Commutative Algebra, Exercise 8.5

The exercise asks the reader to prove that $X$ is a finite covering (i.e., the number of points of $X$ lying over a given point of $L$ is finite and bounded) of $L$, where the affine varieties $X$ and ...
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1answer
64 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
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1answer
36 views

Support of a quasicoherent sheaf

When $M$ is a finitely generated module over a commutative ring $R$, it is easy to see that the support of $\tilde{M}$ on $\mathrm{Spec}\,R$ is given by $V(\mathrm{ann}_R(M))$. This is not true for ...
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1answer
61 views

Help with a problem from Christian Peskine's book about Artinian rings

I am stuck with this problem from the book of Complex Projective Geometry. Let $A$ be a Noetherian ring. Assume that if $a \in A$ is neither invertible nor nilpotent, then there exist $b \in A$ such ...
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2answers
82 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring \begin{equation} S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.\end{equation} Show ...
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31 views

Integral dependence of coordinate ring

In Hartshorne P18-P19, the proof of Thm. 3.4 shows that the ring $S(Y)_{(x_{i})}$ is contained in the integral closure of the coordinate ring $S(Y)$ (all regarded as subrings of the quotient field of ...
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61 views

Direct product of direct sum of a flat module

I have a problem concerning flat modules: Let $M$ be an $R$-module such that the direct product $M^A$ is flat for all sets $A$. I want to prove that $(M^{(B)})^A$ is also flat for any sets $A$ ...
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0answers
27 views

Krull dimension $A/P$ where $A=\mathbb{C}[x,y,z]/(xy,xz)$ [closed]

Krull dimension $A/P$ where $A=\mathbb{C}[x,y,z]/(xy,xz)$ $a)\ P=(\overline{x})$ $b)\ P=(\overline{y},\overline{z} )$ Is it true $\dim(A/P)=\dim (\mathbb{C}[x,y,z]/(xy,xz,y,z))$? and I need ...
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1answer
44 views

Irreducible components in the spectrum of a ring

I have a question concerning page 43 of this book. In Corollary 2.7 it says that the map $\mathfrak{p}\mapsto \overline{\{\mathfrak{p}\}}$ is a bijection from Spec($A$) onto the sets of closed ...
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2answers
52 views

Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$.

Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$. The problem says first verify $p_1=(x,y)$, $p_2=(x,z)$ and $p_3=(x^2-y,z-1)$ are prime minimal over $I$. How can I use it ?
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2answers
61 views

Help with $\sqrt{I}$, where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$

$a)$ $\sqrt{I}$ where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$. first it's clear $y \in \sqrt{I}$ then $x=(x+yz)-yz \in \sqrt{I}$ because $yz \in \sqrt{I}$ is it $\sqrt{I}=(x,y)$ ? $b)$ ...
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1answer
35 views

Find $\operatorname{ht}(p)$, $p=(x_n-x_1^n,\ldots ,x_2-x_1^n)$ ideal of $\mathbb{C}[x_1,\ldots,x_n]$

Find $\operatorname{ht}(p)$ where $p=(x_n-x_1^n,\dots,x_2-x_1^n)$ ideal of $\mathbb{C}[x_1,\ldots,x_n]$ $\operatorname{ht}(p)=$ height of a prime $p$ how to prove $p$ is prime ?
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1answer
48 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
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1answer
29 views

Noether normalisation $A=\mathbb{C}[x,y]/(f)$ where $f=(x-a)y^2-(x-b)$ find a transcendence element

Noether normalisation $A=\mathbb{C}[x,y]/(f)$ where $f=(x-a)y^2-(x-a)$ $a , b \in \mathbb{C}$ find $z \in A$. transcendence over $\mathbb{C}$ such that $A$ is integral over $\mathbb{C}[z]$ any ...
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2answers
110 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
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0answers
136 views

If $A$ is complete for $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology

If $A$ is complete for both $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology. (Matsumura, CRT, Exercise 8.1) How can I solve this problem? A is a ring ...
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1answer
70 views

Prove the ideal $(f)$ is not maximal

I'm trying to solve the following problem: Let $B$ be a UFD and $A := B[y]$ the polynomial ring. Let $f$ be a polynomial that has a term $by^i$ with $i > 0$ such that $b$ is not divisible ...
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83 views

Is $\operatorname{depth}(I,M)=\operatorname{depth}(S^{-1} I,S^{-1} M)?$ [closed]

Let $A$ be a ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module such that $M\neq IM$. Show that there is a maximal ideal $\mathfrak m$ of $A$ such that ...
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0answers
23 views

$ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ [duplicate]

Let $ A $ and $ B $ be two commutative rings. Why is : $ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ ?. Thanks a lot.
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79 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
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1answer
45 views

Transcendence degree of fraction field

Let $k$ be a field and $p \in k[x_1, \dots, x_n]$ an irreducible element. Is there an elementary way to prove that $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?
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0answers
42 views

A question about the proof of Hilbert's Basis Theorem

I have a question regarding the proof of Hilbert's Basis Theorem. Say $I=(f_1,f_2,f_3,\dots)$ is an ideal in $A[x]$, where A is a Noetherian ring. Say we take the leading coefficients $a_i$ of all ...
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1answer
30 views

Integral dependence and field extension

Let $R$ be a domain (commutative with unity). $k$ is field algebraically dependent on $k_0$. $A$ is some ideal of $R \otimes_{k_0} k$ and $A_0$ = $A \cap R$. How to prove that $(R \otimes_{k_0} k)/A$ ...
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1answer
53 views

A question related to associated prime ideals

Let $f:A\to B$ be a (commutative) ring homomorphism, $f^*:\operatorname{Spec}A\leftarrow\operatorname{Spec}B$ the induced map, and $N$ a $B$-module. It is well known that ...
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1answer
44 views

Characterization of Discrete Valuation Rings

Let $R$ be a Noetherian local domain with unique maximal ideal $M$. Then I want to show that if every $M$-primary ideal is a power of $M$, then $R$ is a Discrete Valuation Ring. I know I'll be ...
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1answer
35 views

Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?

I have that $R$ is a Noetherian local ring with maximal ideal $M$, and I want to show that $M/M^2$ is a finite dimensional vector space over the field $R/M$. I think I've proved this (though I ...
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1answer
107 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 ...
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2answers
41 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
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2answers
81 views

What kind of algebraic structure is this

I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with ...
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1answer
56 views

Prove that if the induced homomorphism $M/\mathfrak aM \to N/\mathfrak aN$ is surjective, then $f$ it's surjective.

This problem it's from Atiyah and Macdonald, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ contained in the Jacobson radical. Let $M$ be an ...
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2answers
195 views

What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?

Let $k$ be a field. Let $A$ and $B$ be two $k$-algebras, ie. two rings that are also $k$-vector spaces and their multiplication is $k$-bilinear. Any isomorphism of $k$-algebras is also a ring ...
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1answer
30 views

One dimensional noetherian domain

Let $(R,m)$ be a one-dimensional Noetherian domain. Is $R$ a regular or a topical ring like Gorenstein or other kinds?
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1answer
17 views

Finding a particular principal open subset of $Spec R$

Let $V\subseteq U$ be open subsets of $X=\text{Spec } R$, where $R$ is a commutative ring. So $V$ is the set of prime ideals not containing some ideal $I$, and $U$ is the set of prime ideals not ...
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2answers
28 views

Residue field of a local ring as field extension

Let $k$ be a field, $A$ a finitely generated, commutative $k$-Algebra and $\mathfrak p$ a prime ideal of $A$. Let $K$ be the residue field of the local ring $A_\mathfrak{p}$. I want to show that $K$ ...
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1answer
39 views

Question about completion of DVR.

Let $(R, (\pi))$ be a discrete valuation ring with residue class field $R/(\pi) \cong k$. It is well known that if $k$ embedds into $R$, then there is an isomorphism of the completion $\hat{R} \cong k ...
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1answer
35 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...