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

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Every prime is maximal in a Jacobson ring?

In Attiyah commutative algebra page 71, it is given some equivalent definitions of Jacobson ring. One of the definitions are that every prime ideal which is not maximal is equal to the intersection of ...
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2answers
83 views

There's no surjective ring homomorphism from $\mathbb{Z}[x_1,\dots,x_n]$ onto $\mathbb{Q}$.

I'm trying to prove that, if a field $A$ is also a finitely generated $\mathbb{Z}$-algebra, then $A$ is finite. The proof I found for this depends on the fact that $\mathbb{Q}$ cannot be a finitely ...
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39 views

Finding integral dependence for polynomials [closed]

How can I find a integral dependence for each $f\in K[x]$ over $K[x^2]$? For arbitrary field $K$.
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1answer
36 views

A noetherian local ring having a height one principal prime is a domain

$A$ is a commutative ring with with $1$. If $A$ is a Noetherian and local ring and $A$ has a principal prime ideal of height $1$ then show that $A$ is a domain. Can anybody give some hint.I tried ...
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43 views

Locally free sheaf on Cohen-Macaulay scheme and Serre's criterion

Let $X$ be a projective locally Cohen-Macaulay scheme and $\mathcal{F}$ be a locally free sheaf on $X$. If I understand correctly the definition of Serre's criterion $S_k$, $\mathcal{F}$ satisifies ...
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1answer
73 views

Is $k[x][[h]]$ finitely generated as $k[[h]]$-algebra?

Is $k[x][[h]]$ finitely generated as an algebra over $k[[h]]$, where $k$ is a field, and $xh=hx$.
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1answer
54 views

Open embedding and localization.

Let $X, Y$ be algebraic varieties. If we have an open embedding $X \hookrightarrow Y$, then we have a map $\mathbb{C}[Y] \to \mathbb{C}[X]$. Is $\mathbb{C}[X]$ a localization of $\mathbb{C}[Y]$? For ...
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On the existence of finitely generated modules with finite injective dimension

Assume $R$ is a commutative local Noetherian ring. It is known that if there is a finitely generated module with finite injective dimension then $R$ is Cohen-Macaulay. My question is: if $R$ is ...
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56 views

Geometric interpretation of Ideals in a Prime ideal

I have been told this has a geometric meaning, $I_j \in F[x_1,...x_n]$ be ideals such that $\cap_1 ^n I_J= P$ for P been a prime ideal, then we know that $P= I_j$ for some j=1,..n My Understanding I ...
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1answer
76 views

Is a surjective $R$-endomorphism over a finitely generated $R$ algebra always bijective?

Let $R$ be a unital commutative ring and $A$ a finitely generated $R$-algebra. I found out that if $R$ is a field, then any surjective $R$-endomorphism over $A$ must be injective, too. Does that hold ...
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37 views

relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
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129 views

Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$?

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
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1answer
60 views

Krull dimension of generic fiber

Let $p$ be a prime and $A=\mathbb{Z}_p[t_1,\ldots,t_n]/I$ a reduced flat and irreducible $\mathbb{Z}_p$-algebra of finite type and of Krull dimension d. Let $e$ be the Krull dimension of the ...
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1answer
98 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
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1answer
58 views

Is a prime principal ideal which is not maximal among principal ideals always idempotent?

Let $R$ be a commutative ring with identity, $P$ a prime principal ideal of $R$. Suppose that there exists a proper principal ideal $I$ of $R$ which is strictly larger than $P$ (i.e. $R\supsetneq ...
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1answer
55 views

Surjective morphism of varieties with finite fibers but not “finite”

Let $X$ and $Y$ be affine varieties, and $f : X \to Y$ a dominant regular map. Following Shafarevich, I will call $f$ finite if the induced map on coordinate rings is integral. One consequence of ...
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1answer
119 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
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45 views

Spectral sequences and Ext between extension of modules

Suppose $A$ is a commutative ring, $M_1,M_2,N_1,N_2$ are $A$-modules and we have two exact sequences of $A$-modules $$0\to M_1\to M\to M_2\to 0,$$ $$0\to N_1\to N\to N_2\to 0.$$ I want to write a ...
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36 views

Tensor product of flat modules - proof verification

Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras. Let $M$ be a flat $B$-module and $N$ a flat $C$-module. I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module. ...
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2answers
53 views

What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ ...
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50 views

Determine whether a regular surjective map is finite

Consider the regular map between affine closed sets $f \colon \mathbb{A}^1 \rightarrow \mathcal{Z}(y^2-x^3) \subseteq \mathbb{A}^2$ given by $f(t) = (t^2,t^3)$. $f$ is obviously a dominant map. I ...
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1answer
68 views

Does $(a)=(b)$ imply that $a$ and $b$ are associate in a principal ideal ring?

Let $R$ be a commutative principal ideal ring with identity. Suppose that $a,b\in R$ and $(a)=(b)$. I'd like to know if there always exists $u\in R^\times$ such that $a=bu$. I know several ...
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61 views

Atiyah-MacDonald, 3.18. Why is $B_q$ a local ring of $B_p$?

This question is on the hint that the book gives to finish the exercise. Namely, if $f: A \rightarrow B$ a flat homomorphism of rings, $q$ a prime ideal of $B$ and $p = q^c$, then $B_q$ is a local ...
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1answer
65 views

Projective modules over Dedekind Domains

Show that if $R$ is a Dedekind domain, then every projective $R$-module (not necessarily finitely generated) is a direct sum of ideals of $R$. I have spent a while on this problem and I wonder if it ...
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1answer
46 views

Properties of module length

Let $e_{A}(\phi, M): = l_A(\mathrm{coker}(\phi) ) - l_A(\ker(\phi))$. In my book it is stated that if $IM = 0 \implies e_{A}(\phi, M) = e_{A/I}(\phi, M)$ and this seems to be obvious for the author. ...
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93 views

What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in ...
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1answer
69 views

Irreducible components of schemes

Consider the scheme $X:=\mathrm{Spec}(k[X,Y]/(X^2,XY))$. According to Qing Liu's "Algebraic geometry and arithmetic curves", the irreducible components are in $1-1$ correspondence with subschemes of ...
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43 views

Associated graded ring of a quotient

Given a ring A and an ideal $I \subseteq A$ we can form its associated graded ring with respect to $I$ $$ Gr_I(A)= A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots $$ I wondered if there is a way to ...
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75 views

Must a $R$-automorphism on $R[X]$ be of the form $X\mapsto aX+b,\ a\in R^*,b\in R$?

Let $R$ be a commutative ring. I wonder if every $R$-automorphism (that is, a ring automorphism that fix $R$) $\varphi$ of $R[X]$ satisfies $\varphi(X)=aX+b$, where $a$ is an unit in $R$ and $b$ an ...
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2answers
92 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
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Equivalence of definitions for completion

For the settings on my question, take Atiyah's chapter on completions. Basically we have two definitions of completness (Atiyah's sense, the canonical map $\phi:M\rightarrow \widehat{M}$ is an ...
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1answer
31 views

An example of a (necessarily non-Noetherian) ring $R$ such that $\dim R[T]>\dim R+1$

What is an example of a non-Noetherian ring $R$ such that the Krull dimension of $R[T]$ is greater than dim$R+1$?
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How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite? I'm looking for equivalent ...
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Prime ideals contained in the union of almost all prime ideals

I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is ...
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31 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
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1answer
56 views

Proving a ring is Noetherian when all maximal ideals are principal generated by idempotents

Let $R$ be a commutative ring with unity such that all maximal ideals are of the form $(r)$ where $r\in R$ and $r^2=r$. I wish to show that $R$ is Noetherian. I know that if all prime (or ...
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1answer
47 views

Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping.

Let $f:A \rightarrow B$ be an integral homomorphism of rings. Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping. My Try: So, $B$ is integral over $f(A)$. $f^*$ is given by $b\longmapsto ...
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77 views

How to imagine the difference between the following schemes?

Consider $A=\operatorname{Spec} k[x]_{(x)}[t]$ and $B=\operatorname{Spec} k[x,t]_{(x)}$ for a field $k$ (Vakil, note 11.3.8). For me, both are infinitesimal neighborhoods of an affine line - the ...
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1answer
77 views

Finding the kernel of $\alpha: K[X,Y,Z]^{3}\rightarrow \langle X,Y,Z\rangle$, $(f,g,h)\mapsto Xf+Yg+Zh$.

I am trying to do exercise $2.3$ of Reid's "Undergraduate Commutative Algebra": Let $A=K[X,Y,Z]$ where $K$ is a field, and $m=\langle X,Y,Z\rangle$. I have to show that the kernel of the ...
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1answer
48 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
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If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
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1answer
35 views

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$?

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$? Here $E(M)$ is the injective envelope of $M$ and $\operatorname{Ass}$ denotes the set ...
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2answers
49 views

Dimension of Tensor Product for Flat Extensions

Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map. I am trying to determine whether $$ \dim B\otimes_AC=\dim B+\dim C-\dim A $$ Any counterexamples ...
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1answer
47 views

Determine whether $(\mathbb{R}[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is a discrete valuation ring.

Geometrically, the curve $y^2-x^2-x^3=0$ is singular at the origin in the real plane. Thus the ring should not be a dvr. I am thinking to show that it is not a dvr, it is equivalent to show that it ...
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37 views

Polynomial ring with integral coefficients is integral

Let $B$ be a ring and $A\subset B$ a subring. Assume that $B$ is integral over $A$. I have to prove that $B[X]$ is integral over $A[X]$. I tried writing down an integral relation for $f(X)\in B[X]$ ...
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2answers
83 views

If local rings are Noetherian, is scheme locally-Noetherian?

If all the local rings of a scheme are Noetherian, is the scheme locally-Noetherian?
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1answer
18 views

A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
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57 views

Kernel identity of the canonical homomorphism

Let $A$ be a commutative complete Noetherian ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $M_0$ is finitely generated over $A$. Let $(M_n)_{n\geq 0}$ be ...
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121 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
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34 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...