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

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Matrix of linear forms

I would like to see if the following is true: Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field. Let $M$ be a $(a+n-1)\times a$ matrix of linear forms in $R$. If $I_a(M)$ is $(x_1,\ldots, ...
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0answers
27 views

Reference for the determinant of an endomorphism of a projective module of finite rank

In Bourbaki algèbre commutative first book exercice 9 of paragraph 5 of chapter II (page 174) there is an exercise where they explain how to define the determinant of an endomorphism of a projective ...
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1answer
41 views

Show that $\alpha_A^{-1}(I'+J')=\alpha_A^{-1}(I')+\alpha_A^{-1}(J')$, where $I',J'$ are ideals and $\alpha_A$ is a surjective ring homomorphism.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ constant matrix. Let $I',J'$ be ideals in $k[y_1,...,y_n]$. ...
4
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0answers
77 views

When flatness of $B$ over $A$ implies flatness of $B$ over $C$, where $A \subseteq C \subseteq B$?

Assume $A \subseteq C \subseteq B$ are integral domains, with $B$ flat over $A$. Generally, $B$ is not necessarily flat over $C$. For example, see van den Essen's book "Polynomial Automorphisms and ...
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1answer
27 views

Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \left\langle\alpha_A(I)\right\rangle \cap \left\langle\alpha_A(J)\right\rangle $.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ matrix. Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset ...
3
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1answer
121 views

Projectivity of $B$ over $C$, given $A \subset C \subset B$

I have found a result concerning projectivity of a certain ring extension: Lemma 2.64. This says the following: Let $A$ be an integral domain or a noetherian ring, $B$ an $A$-algebra, $C$ an ...
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1answer
45 views

Krull dimension of $A[x]/\langle x^2 + 1 \rangle$

Consider any noetherian ring $A$ and the polynomial ring $A[x]$. Consider the quotient ring $A[x]/\langle x^2+1\rangle$. Is the dimension of this quotient ring equal to dimension of $A$ (i.e. ...
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1answer
45 views

How can one show that an ideal with some property is zero-dimensional?

Let $\mathfrak{a}$ be an ideal in $\mathbb{k}[x_1, \ldots, x_n]$ and a Gröbner basis of the ideal be $\{g_1, \ldots, g_t\}$. For each $i = 1, \ldots,n$, there exists $j \in \{1, \ldots, t\}$ such that ...
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1answer
57 views

Why is the affine $\Bbbk$-algebra, $ \Bbbk[x]/\langle x^3 \rangle $ zero-dimensional?

Consider the ideal $\mathfrak{a} = \langle x^3 \rangle \subseteq \Bbbk[x]$. The ideal $\langle x + \mathfrak{a} \rangle$ is a prime ideal in $ \Bbbk[x]/\mathfrak{a}$. Then why is the affine algebra, ...
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1answer
29 views

Separability implies flatness, in a special case

A nice theorem of Wang, Corollary 9 of A Jacobian criterion for separability, says the following: Let $B=A[z]=A[Z]/(h(Z))$. If $B$ is a separable algebra over $A$, then $B$ is a flat module over ...
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1answer
41 views

Ring localization and ideals

I'm trying to solve a couple of problems involving ring localization and I'm not sure if my solutions are right or if I understand the idea of localization correctly. Let $A$ be a commutative ...
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0answers
28 views

Clarification on notation in Siegfried Bosch's Commutative Algebra book about primary decomposition of ideals.

I'm reading through Siegfried Bosch's Commutative Algebra book, and I'm confused on his notation in one his proofs. He uses this notation a lot, so I think I should I understand it. The notation first ...
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1answer
42 views

System of parameters for a local ring

I need some help to solve this problem. This is the kind of problem that makes me stuck at the very beginning. Let $K$ be algebraically closed, $X = \{(x,y)\in\mathbb{A}^2_K: \ y^2-x^3=0\}$ an ...
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1answer
51 views

Flatness and normality

I have just read: Direct proof of non-flatness and wondered what is exactly the claim that Alex Youcis is referring to: "...but are you aware of the fact that flatness preserves normality. In your ...
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1answer
18 views

Extending an absolute value over a localization.

Let's consider the definition of (algebraic) absolute value given by Wikipedia (https://en.wikipedia.org/wiki/Absolute_value_%28algebra%29), and focus the attention under the voice "Fields and ...
2
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1answer
36 views

Injective map from integral domain to integrally closed domain? [closed]

If you have an injective map from an integral domain to an integrally closed domain, is that necessarily an integral extension? If so, is there an induced injective map on the respective field of ...
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1answer
47 views

Integral extension of local ring

I suppose this is a classical result, but I'm having problems to prove it. I want to prove that if $R$ is a commutative local ring and $R\subset S$ is an integral extension, then $S$ is also ...
5
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2answers
205 views

Cohen-Macaulay but not regular

In the Wiki page it is claimed that $K[[t^2,t^3]]$ is a $1$-dimensional Cohen-Macaulay ring which is not regular. Is there anybody who kindly explain to me the above assertion? Thanks in ...
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1answer
40 views

Class number and complex conjugation

Let $h$ be the be the class number of the ring of integers of the $p$th cyclotomic field. Suppose $p\mid h$ and let $I$ be an ideal of order $m$ such that $p \mid m$. Does $p$ divide the order of $I ...
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1answer
45 views

When $f(I)S=S$ for each ideal $I$ of $R$?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). Question 1. ...
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1answer
25 views

Prime ideals in a Dedekind domain

If $R$ is a Dedekind domain and $I\subset R$ is a non-zero ideal then by the Noetherian property of $R$, I can show that there are distinct non-zero prime ideals $P_1,...,P_r$ s.t. $P_1^{a_1}\cdots ...
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1answer
24 views

Localization of ideals at all primes

Let $R$ be a commutative ring with $1$ and $I$, $J$ ideals in $R$. For a prime ideal $P$, let $I_P=(R-P)^{-1}I$ be the localization of $I$ at $P$. Question: If $I_P=J_P$ for all prime ideals ...
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3answers
120 views

In a reduced ring the set of zero divisors equals the union of minimal prime ideals.

If $R$ is a reduced commutative ring with identity, why is the set $Z$ of zero divisors the union of minimal prime ideals? I know that $Z$ is a union of associated primes, and that the ...
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3answers
79 views

If the intersection of ideals $I_{1},\ldots,I_{n}$ is contained in a prime ideal $P$, then one of them is contained in $P$

Let $A$ be a commutative ring and $I_{1},\ldots, I_{n}$ and $P$ ideals in $A$ with $P$ prime so that $\cap_{i=1} ^{n} I_{i} \subset P $. Show that there's an $i_0 \in \{1,...,n \}$ so that $I_{i_0} ...
2
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1answer
40 views

Equivalence relation on regular functions

In this problem, consider $K$ an algebraic closed field and $X\subset\mathbb{A}^n_k$ an irreducible variety. Given an open Zariski $U\subset X$, we say that a function $\phi:U\rightarrow K$ is regular ...
2
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1answer
30 views

Height and minimal number of generators of an ideal.

Can anyone could give me a reference in a book about the proof of the following Let $I$ be an ideal of a ring. We denote with $\operatorname{ht}(I)$ the height of $I$, and by $\mu(I)$ the minimal ...
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1answer
22 views

Exactness of Dual Sequence, A Proposition in Atiyah and MacDonald

The proposition 2.9 of Atiyah and Macdonald syas that a sequence of $A$-modules $$M'\xrightarrow u M \xrightarrow v M'' \rightarrow 0$$ is exact iff the dual sequence $$0\rightarrow Hom ...
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1answer
23 views

Infinite direct product of C-M rings

A finite direct product of Cohen-Macaulay rings is a Cohen-Macaulay ring. It could be checked by a scrutiny into localization of a finite direct product of rings at a prime ideal of the product. Now, ...
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1answer
90 views

Flatness of $\Omega_{B/K}$ over $B$.

Let $K$ be a field of characteristic zero. Assume that $K \subset A \subseteq B$ are noetherian integral domains, with $A$ regular (= all its localizations at maximal ideals are regular local rings). ...
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1answer
67 views

Showing there are at least one and only finitely many maximal ideals containing the extension of a maximal ideal [closed]

Let $F$ be a field and $M$ a maximal ideal of $F[x_1, x_2, ..., x_n]$. Let $K$ be an algebraic closure of $F$. Show that $M$ is contained in at least one and in only finitely many maximal ideals of ...
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2answers
111 views

Is a specific ring extension $B$ of $K[x,y]$ integrally closed? separable?

Let $A=K[x,y] \subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
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1answer
89 views

Surjection $M\to R/P$

Let $(R,\mathfrak{m})$ be a local Noetherian ring, and $M$ a finitely generated $R$-module. I am trying to show that there is a surjection $M\to R/P$ for any $P\in\operatorname{Supp} M$. I know ...
5
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46 views

Geometric statement of Prime Avoidance?

The Prime Avoidance Theorem is very clean to state in algebraic terms: Let $I \subset R$ be an ideal (with $R$ noetherian) and $I \subseteq \bigcup_{i=1}^r P_i$, where each $P_i$ is prime. Then $I ...
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1answer
32 views

Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like.

Let $S=\bigoplus_{n=0}^\infty S_n$ be a graded ring. We denote $S_+=\bigoplus_{n>0}^\infty S_n$. As usual we define $\text{Proj}(S)$ to be the set of homogeneous, prime ideals $\mathfrak p$ of $S$ ...
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32 views

Question on complete intersection ideal.

Let $R$ be a Noetherian commutative ring with unity and let $I$ be an ideal of $R$. Suppose I want to know if $I$ is a complete intersection, I know that $I$ is finitely generated but I am unable to ...
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2answers
35 views

Does $I(J\cap K)=IJ\cap IK$ hold in a finitely generated polynomial $K$-algebra for $K$ a field?

Let $K$ be a field and $R:=K[X_1,X_2,\cdots, X_n]$ for a certain $n\in\mathbb N$. If $I,J,K$ are three ideals of $R$, can we conclude that $I(J\cap K)=IJ\cap IK$?
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2answers
84 views

When regularity of $A$ implies regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
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1answer
39 views

Explicit description of the inverse image sheaf of an ideal sheaf.

$\DeclareMathOperator{\Spec}{Spec}$ Let $f: \Spec A \to \Spec B$ be a morphism of affine schemes and $f^\#: B \to A$ be the corresponding ring homomorphism. Let $\mathcal{I} \subseteq ...
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0answers
32 views

Showing function defined on $\text{Frac}(R)$ is a ring homomorphism

Let $f : R \to S$ be a ring homomorphism where $R, S$ are integral domains. I want to show that $\varphi : \text{Frac}(R) \to \text{Frac}(S)$ defined by $r/1 \mapsto f(r)/1$ is a ring homomorphism. ...
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1answer
82 views

Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?

Fix a positive integer $p$, possibly prime. For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow ...
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1answer
31 views

Height one prime ideals in Cohen-Macaulay integral domains

Is it true that a height one prime ideal in a Cohen-Macaulay integral domain $R$ is principal? Is the corresponding quotient domain Cohen-Macaulay? My think is that the grade of the prime ideal ...
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1answer
84 views

What is the kernel of $R[T] \to R[w]$, $T \mapsto w$, $w=u/v$, $u,v \in R$, where $R$ is an integrally closed domain?

I am posting the following question after posting a similar question: What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$? If $R$ is an integral domain, $w=u/v$, where $u,v ...
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14 views

$SL(2)$ invariant polynomials are generated by determinant?

Suppose $SL_2(\mathbf{C})$ acts on the space of quadratics $aX^2+2bXY+cY^2$ by $X\to \alpha X+\beta Y, Y\to\gamma X+\delta Y$, where $\alpha,\beta,\gamma,\delta$ consists a matrix in ...
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1answer
86 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
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0answers
57 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
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1answer
36 views

Is $K[[x]]$ an Artinian/Noetherian $K[x]$-module?

Let $K$ be a field an consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian. I used the following propositions: If M is an $R$-module and $N\subseteq M$ a submodule, then ...
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0answers
25 views

Difference between division algorithm and buchberger's algorithm

what is the main difference between division algorithm and buchberger's algorithm? I think we use the same steps where non-zero remainder is said to be the new polynomial and continue this process ...
6
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3answers
85 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
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0answers
39 views

Better understanding regular functions on a Projective variety

Hi guys I was just looking an example from class that was left as obvious, but it is not so obvious to me. $W= V(x_1x_4-x_2x_3)= $ where $I(W)= \langle x_1x_4-x_2x_3 \rangle$ so we just picked an ...
5
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1answer
43 views

Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$?

Let $R \subseteq S$ be commutative rings. $S$ is separable over $R$ if $S$ is a projective $S \otimes_R S$-module (under $\mu: S \otimes_R S \to S$ defined by $\mu(s_1 \otimes s_2)=s_1s_2$). Let ...