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

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1answer
62 views

Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence?

I have just started reading about the concept of $M$-regular sequences on my own and to understand the definition I asked myself the following question: Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence? ...
12
votes
5answers
322 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
1
vote
1answer
74 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
1
vote
1answer
117 views

Flatness of module over field of fractions

This is from Liu 1.2.9. Let $A$ be an integral domain, and $K$ its field of fractions. Let $M$ be a finitely generated sub-$A$-module of $K$. Why do $M$ is flat if and only if $M_{\mathfrak p}$ is ...
1
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1answer
72 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
3
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1answer
113 views

Let $f: U \rightarrow W$ be a morphism of affine algebraic sets and $f': k[W] \rightarrow k[U]$ be the k-algebra morphism of coordinate rings.

Prove if $f'$ is surjective then $f$ is a homeomorphism of $U$ onto the closed subset $W$. Well, it's the first time I've seen this word "homeomorphism" but I read online that a map is a ...
1
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0answers
22 views

Direct sum of ideals over Dedekind domain [duplicate]

I'm trying to show that Let $\frak{a},\frak{b}$ be two ideals of a Dedekind domain $\cal{O}$. Show that there is an isomorphism \begin{equation*} \frak{a}\oplus\frak{b}\cong\cal{O}\oplus\...
0
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1answer
67 views

Canonical homomorphism and free module, Liu 1.2.8 c

How can I do the problem 1.2.8 c in "Algebraic Geometry and Aritmetic Curves". Namely, let $A$ be a Noetherian ring, $M$ a finitely generated $A$-module, and $N$ an $A$-module. Let $B$ be a flat $A$-...
1
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2answers
171 views

Total ring of fractions of a Noetherian reduced ring is artinian

I'm doing the preparation to an exam, and I'm stuck in the following: If $R$ is a Noetherian ring with zero nilradical ($N(R) = 0$), and $S$ is the set of regular elements of $R$ ($r \in S$ if $rs ...
4
votes
2answers
191 views

integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maximal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal. 2) and that for every ...
2
votes
1answer
80 views

How to prove that a ring is not flat over $k[t,s]$? [duplicate]

Let $k$ be a field, $A=k[t,s]$, and $C=A[z]/(tz-s)$. How can I prove, using the ideals $tA$ and $sA$, that $C$ is not flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.6(c).) ...
2
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1answer
272 views

Example of an integral domain that is not integrally closed and having some localization which is also not integrally closed [closed]

Can anyone show an example of integral domain that is not integrally closed and also has one of its localization with respect to a maximal ideal not integrally closed?
3
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1answer
69 views

Flatness and intersection of ideals [closed]

This is Exercise 1.2.6(a) in Liu, Algebraic Geometry and Arithmetic Curves Let $B$ be a flat $A$-algebra. Show that for any finite family $\{I_\lambda\}_{\lambda\in \Lambda}$ of ideals of $A$, ...
0
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1answer
68 views

$H^i_I(M)$ is finitely generated iff the support of $Ext^{d-i}_S(M, S)$ has dimension zero

$(R,m)$ is a local Noetherian ring. $M$ is a finite $R$-module. Here, using dualizing complex, Karl Schwede says that if $R=S/I$ where $S$ is regular of dimension $d$, then we have: "$H^i_m(M)$ is ...
0
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0answers
86 views

How cannot localization of any integral domain respect to maximal ideal not be integrally closed?

Suppose that there is integral domain $I$. Now we take localization $I_m$ of $I$ respect to its maximal ideal $m$. $I_m$'s elements will consist of $a/b$ where $a \in I$ and $b \in m$. But integral ...
1
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0answers
125 views

How to check if a polynomial is inside an ideal using a Groebner basis

I'm given that an ideal $I=\langle F_1, F_2, F_3, F_4, F_5, F_6, F_7\rangle$ $F_1=a+b+c-d-e-f$ $F_2=a+b+c-g-h-i$ $F_3=a+b+c-g-e-c$ $F_4=a+b+c-a-e-i$ $F_5=a+d+g-a-e-i$ $F_6=a+d+g-c-f-i$ $F_7=a+d+...
0
votes
2answers
295 views

Noetherian ring with infinite Krull dimension.

I just started to read about the Krull dimension (definition and basic theory), at first when I thought about the Krull dimension of a noetherian ring my idea was that it must be finite, however this ...
3
votes
1answer
83 views

Showing that for every monomial $x^u\in\operatorname{in}_{<}(I)$, there exists $f\in I$ s.t. $\operatorname{in}_<(f)=x^u$

Given an ideal $I\subset R=K[x_1, ...,x_n]$ and let $<$ be a term order on the ring $R$. I must show that $\forall x^u\in\operatorname{in}_<(I)$, $\exists f\in I$ s.t. $\operatorname{in}_<(f)=...
0
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1answer
40 views

Tensor product and localization

This is from Liu, problem 1.2.2. Let $\rho:A\to B$ be a ring homomorphism, $S$ a multiplicative subset of $A$, and $T=\rho (S)$. Show that $T^{-1}B\simeq B\otimes_AS^{-1}A$ as $A$-algebras. I ...
0
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1answer
126 views

Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes

I am reading the proof by Götz-Wedhorn Algebraic Geometry I Theorem 6.45, and also Liu, Theorem 1.14. One thing that I do not understand is this: For easier cases, we assume $X=\text{Spec A}$ and let ...
1
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1answer
83 views

Question about split monomorphisms of free modules over local rings

In May's notes on Cohen-Macaulay and Regular Local Rings, during the proof of Serre's theorem on page 9, he claims that if $R$ is a local ring and $\phi\colon F\to F'$ is a map of finitely ...
2
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1answer
90 views

Maximal ideals of $R[x_1,\ldots,x_n]$ that is $R$ is a commutative rings with identity

Let $R$ be a commutative ring with identity and $R[x_1,\ldots,x_n]$ a polynomial ring over $R$. What are maximal ideals in $R[x_1,\ldots,x_n]$? How are, if $R$ is a Hilbert ring (Jacobson ring)?
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1answer
33 views

Canonical homomorphism related to ideal is an isomorphism

I have a problem to do the exercise 1.2.1 b on Liu. Namely, Let $M$ be an $A$-module, $I\subseteq \operatorname{Ann}(M)$ an ideal, $N\ne M$ is an $A$-module such that $I\subseteq \operatorname{Ann}(N)...
2
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1answer
123 views

Polynomial ring, prime ideal, factor ring

I want to prove that this ideal: $I=(y^3-xz, xy^2-z^2, x^2-yz)$ is prime in $K[x,y,z]$. I think it would be a good idea to prove that the factor ring $K[x,y,z]/I$ has no zero divisors. In this factor ...
1
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1answer
64 views

Lemma for the Krull-Akizuki Theorem

This is from Matsumura's Commutative Ring Theory (Lemma for Theorem 11.7) Lemma for the Krull-Akizuki Theorem Let $A$ and $K$ be as in the theorem, and let $M$ be a torision-free $A$-module of ...
12
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1answer
494 views

What's the “real” reason a finite map has finite fibers?

This is a soft question. I have encountered two very different proofs of what seems like "basically the same theorem," and I want to understand how they relate and "what the real explanation is." ...
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1answer
112 views

Is the mentioned basis a Gröbner basis?

It's mentioned into my notes that if the ideal given as $I=\langle x+y+z, 3x-2y\rangle$, then $\{x+y+z, 5y+3z\}$ is a Gröbner basis for the ideal. I can see how $I=\langle x+y+z, 3x-2y\rangle=\langle ...
2
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3answers
96 views

Exact sequence with flat module tensored by module stays exact

The following theorem is given in Liu proposition 1.2.6: Let $A$ be a ring. Let $0\to M^\prime\to M\to M^{\prime\prime}\to 0$ be an exact sequence of $A$-modules. Let us suppose that $M^{\prime\prime}...
-1
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1answer
110 views

fraction field of polynomial ring that is a finite extension of the base field

Let $k$ be a field. Let $P$ be a prime ideal of $k[x_1, ..., x_n]$. Let $K$ be a field of fractions of $k[x_1, ..., x_n]/P$. Suppose $K$ is a finite extension of $k$. Does it then follow that $P$ is ...
3
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1answer
82 views

“Pushforward” over flat morphisms of functions which are constant on fibers

I believe the following should be true, but I'm not sure where to find the required commutative algebra to prove it: If $\mathrm{Spec}\,A \rightarrow \mathrm{Spec}\,B$ is a flat morphism of algebraic ...
0
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1answer
85 views

An ideal avoidance

It is known that in a commutative ring $R$ an ideal contained in a finite union of prime ideals $P_i , ( i=1,...,n)$ is a subset of one of them (prime avoidance theorem). Now, if $P_i$'s are arbitrary ...
2
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1answer
51 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
3
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3answers
311 views

Regular Ring is Integrally Closed?

Studying some topics in Algebraic Geometry I've bumped into the following question: Let $A$ be a regular ring. Is $A$ integrally closed? Someone said me that with the hypothesis $A$ local ...
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1answer
26 views

$I+J=R$ and $r+s=1, r\in I,s\in J$ then $sx+ry\in IJ\Rightarrow x\in I$ and $y\in J$

Let $R$ be a commutative ring with unity. $I+J=R$ with $I,J$ Ideals and $r+s=1, r\in I,s\in J$ then $sx+ry\in IJ\Rightarrow x\in I$ and $y\in J$. It should be very obvious. How can I conclude that $...
4
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1answer
182 views

Prove that factor modules are isomorphic.

I'm trying to prove (from a previous post) that if $A=k[x,y,z]$ and $I=(x,y)(x,z)$ then $$\dfrac{(x,y)/I}{(x,yz)/I} \cong\dfrac{A}{(x,z)}.$$ I did this by defining the homomorphism $\phi: A \to ((x,...
0
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1answer
185 views

Prime ideals of infinite depth in Noetherian rings

I'm struggling with the definition of depth of prime ideals given in Atiyah's book: The depth of a prime ideal $p$ is longest strictly increasing chain of prime ideals starting at $p$. Clearly $\...
0
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1answer
74 views

An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
3
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5answers
180 views

Why does $p(a)=0$ imply $(x-a) \mid p$?

There's something I've never understood about polynomials. Suppose $p(x) \in \mathbb{R}[x]$ is a real polynomial. Then obviously, $$(x-a) \mid p(x)\, \longrightarrow\, p(a) = 0.$$ The converse of ...
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2answers
72 views

Atiyah-Macdonald p.108

I don't understand the following lines on p.108 (chapter 10) in Atiyah-Macdonald: Since we have a natural homomorphism $f:A\to \hat{A}$ we can regard $\hat{A}$ as an $A$-algebra and so for any $A$-...
0
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1answer
36 views

F structure of an algebraic set (why is this ring hom injective?)

I'm trying to understand the notion of the field of definition of an algebraic set. Specifically, I'm stuck on page 6 of the book Linear Algebraic Groups by TA Springer. Suppose $F \subset K$ is a ...
0
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1answer
129 views

Find a disassembly for a module.

Let $A=k[x,y,z]$ and $I=(x^2,xy,xz,yz)$. My previous question was how to calculate a primary decomposition of $I$. However there was a part b) added to this exercise, namely to calculate a disassembly ...
5
votes
2answers
376 views

Tensor product of two finitely generated modules

How can I show that if $M$ and $N$ are finitely generated $A$-modules, then so is $M\otimes_AN$? I understand that I have assumption that there are integers $n,m$ such that there are surjections $A^n\...
2
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1answer
64 views

$I = (x^2, y^2) ⊂ K[x, y]$; $gin\ (I)=?$

an easy Google search give a lot of results about the definition of generic initial ideal. But all definitions I see, are like this one: I can't use this definition to compute gin(I) even in simple ...
0
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1answer
171 views

Localization of a ring that is not an integral domain

Let $A$ be a commutative ring with unity that is not an integral domain and $\mathcal{P}$ be any prime ideal of $A$. Then I know that $A_{\mathcal{P}}$ is not an integral domain using the ...
0
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0answers
39 views

$I$ is a $J$-primary ideal of $R$ iff $I/L$ is a $J/L$-primary ideal of $R/L$

Let $R$ be a commutative unitary ring and $I$, $J$, $L$ be ideals of $R$ with $L$ proper, $L \subseteq I$ and $L \subseteq J$. A homework question asks to prove that if $R$ is noetherian then $I$ is ...
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1answer
43 views

Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
1
vote
1answer
59 views

The rank of the integral closure as a free module

Let $ O$ be a PID, and let $L$ be a finite separable extension of its quotient field $K$ with degree $n$. Prove that the integral closure of $O$ in the field $L$ is a free module of rank $n$. Here ...
0
votes
1answer
59 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
0
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0answers
37 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
1
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1answer
108 views

What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...