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

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The prime ideals of the ring $K[x]$

I was wondering what the prime ideals of the ring $K[x]$ are, where $K$ is a ring. My guess is that it's any ideal generated by a set of irreducible polynomials over the ring $K$. Have I covered all ...
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89 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a $B$-...
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42 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
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1answer
20 views

Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
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What is the Krull dimension of $\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]$?

I am studying commutative algebra and saw the following question in one of the tests: What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$ I know ...
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70 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to ...
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154 views

What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an ...
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101 views

Quotient of ring is flat gives an identity of ideals

I have problem to understand and solve the exercise 1.2.14 on Qing Liu's book "Algebraic Geometry and Arithmetic Curves". It goes as follows: Let $A\to B$ be a ring homomorphism, and let $J$ be an ...
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94 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j \...
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Integrally closed domain.

Suppose $A$ is a unique factorization domain, $a$ is an element of $A$. Is the ring $A[x,a/x]$ always integrally closed? ($x$ is a variable over $A$) Thanks!
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84 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field $k$ with $\operatorname{char} k\not=2$, and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals $M$ ...
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211 views

Atiyah and Macdonald, exercise 11.7

I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian. The very beginning poses a problem, he states in the hint that: for a prime of height $m$ we can ...
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90 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions $\...
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78 views

how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
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3answers
134 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that $f(X)=Z(\...
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Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; \...
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1answer
72 views

Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
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1answer
112 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
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88 views

Flat algebra over a Dedekind domain

Let $B$ be a flat algebra over a Dedekind domain $A$. Let $f\in B$ be such that for every maximal ideal $\mathfrak m$ of $A$, the image of $f$ in $B/\mathfrak mB$ is not a zero divisor. How can I show ...
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The annihilator numbers of $S/I$

Let $S=K[x_{1},x_{2},...,x_{n}]$ and $I$ be a strongly stable ideal of $S$. Compute the annihilator numbers of $S/I$ with respect to the almost regular sequence $x_{n},x_{n-1},...,x_{1}$. (...
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1answer
70 views

Finding conditions for $\mathbb Z[i][X,Y]/(Y^2 - aX)$, $a \in \mathbb Z[i]$ to be regular

I am trying to find the dimension and the necessary and sufficient conditions under which $A[X,Y]/(Y^2 - aX)$ is regular, that is, the localizations of $A[X,Y]/(Y^2 - aX)$ at all maximal ideals are ...
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107 views

Finding an ideal such that quotient is Cohen-Macaulay

Let $R$ be a commutative local Noetherian ring which is not a domain and not Cohen-Macaulay. Can we find an ideal $I$ in $R$ such that $R/I$ is Cohen-Macaulay, and $\dim R/I=\dim R$?
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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? ...
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5answers
327 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 ...
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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 ...
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1answer
119 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 ...
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73 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. ...
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116 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 ...
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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\...
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1answer
68 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 $A$-...
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195 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 ...
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196 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 ...
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1answer
81 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).) ...
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282 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?
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73 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 $A$, ...
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69 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 ...
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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 ...
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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+...
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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 ...
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84 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)=...
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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 ...
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131 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 ...
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84 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 ...
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92 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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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)...
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125 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 ...
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1answer
65 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 ...
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1answer
506 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 ...
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3answers
98 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}...