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

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2
votes
1answer
35 views

Congruence in localization of rings

Please help me to prove for all maximal ideals $\mathfrak{m}$ of $R$, $(aR/a^2R)_\mathfrak{m}\cong (aR)_\mathfrak{m}/(a^2R)_\mathfrak{m}\cong aR_\mathfrak{m}/a^2R_\mathfrak{m}$, where $R$ is a ...
3
votes
1answer
81 views

Is the unique morphism from the empty scheme $\operatorname{Spec}((0))$ to some other scheme $X$ smooth?

This is a very pedantic question, but Is the unique morphism from the empty scheme $\emptyset = \operatorname{Spec}((0))$ to some other scheme $X$ smooth?
4
votes
1answer
218 views

Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5)

This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows: ...
2
votes
1answer
39 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 ...
3
votes
0answers
35 views

Localization and Direct limit [duplicate]

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 ...
3
votes
0answers
26 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 ...
2
votes
0answers
77 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 ...
0
votes
1answer
63 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.
19
votes
2answers
604 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 ...
2
votes
1answer
65 views

Prime radical that is nil but not nilpotent

Please help me to show that the prime radical of the ring $R=\prod\limits_{n = 1}^\infty { \mathbb{Z} /2^n\mathbb{Z} } $ is nil but not nilpotent.
1
vote
1answer
80 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 ...
2
votes
1answer
138 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ 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 ...
1
vote
1answer
221 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 ...
1
vote
1answer
52 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 ...
3
votes
1answer
104 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) ...
2
votes
1answer
89 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 ...
1
vote
0answers
238 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 ...
3
votes
1answer
90 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 ...
1
vote
1answer
67 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 ...
8
votes
2answers
176 views

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

I am stuck with problem 3, chapter 4, from the book of Peskine, An Algebraic Introduction to Complex Projective Geometry. Let $A$ be a Noetherian ring. Assume that if $a \in A$ is neither ...
7
votes
2answers
381 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$. Show the existence or nonexistence of isomorphism between $S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}$ and $\prod_{1}^{\infty}\mathbb{Q}_{i}$ as $\mathbb{Q}$-vector spaces. ...
0
votes
1answer
189 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 ...
6
votes
3answers
268 views

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed?

Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ integrally closed? (Or could it have a relation to another domain like $\mathbb{Z}[\sqrt{-3}]$ does with $\mathbb{Z}[\omega]$?) Also, is it UFD? What are its ...
-2
votes
1answer
160 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 ?
1
vote
2answers
73 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 $\sqrt{I}=(x,y)$ ? $b)$ ...
0
votes
1answer
51 views

Find the height of prime ideal $p=(x_n-x_1^n,\ldots ,x_2-x_1^n)$ in $\mathbb{C}[x_1,\ldots,x_n]$

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 ?
2
votes
1answer
67 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 ...
0
votes
1answer
38 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 ...
5
votes
2answers
247 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 ...
0
votes
1answer
103 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 ...
5
votes
0answers
188 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 $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
2
votes
1answer
106 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 $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?
3
votes
0answers
75 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 ...
2
votes
1answer
42 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 $(R \otimes_{k_0} k)/A$ ...
-1
votes
1answer
66 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 ...
1
vote
1answer
75 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 ...
4
votes
1answer
93 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 ...
1
vote
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 ...
2
votes
2answers
57 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 ...
2
votes
2answers
100 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 ...
2
votes
1answer
127 views

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

This problem is from Atiyah and Macdonald, Introduction to Commutative Algebra, Exercise 10, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ ...
6
votes
2answers
254 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 ...
-4
votes
1answer
115 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?
0
votes
1answer
32 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 ...
1
vote
2answers
205 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 $K$ ...
1
vote
1answer
105 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 ...
1
vote
1answer
60 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 ...
3
votes
1answer
286 views

Exercise from Kaplansky's Commutative Rings and Eakin-Nagata Theorem

Exercise 15 of section 2-1 of Kaplansky's Commutative Rings is to show that if $T$ is a Noetherian ring and is finitely generated module over a subring $R$ of $T$, then $R$ is Noetherian. Kaplansky ...
2
votes
1answer
68 views

Possible examples where the Zero Divisor Conjecture does not hold

Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$. I was ...
1
vote
1answer
96 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?