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

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2
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0answers
59 views

prove/disprove $\Delta$ is strongly connected.

Let $\Delta$ be a simplicial complex and $F_1,...,F_n$ be the facets of $\Delta$. Let $\Delta_1$ be another simplicial complex and $F_1,...,F_{n-1}$ be the facets of $\Delta_1$. Assume $\Delta$ and ...
1
vote
0answers
43 views

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, ...
1
vote
0answers
28 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 ...
2
votes
1answer
42 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
votes
0answers
81 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 ...
2
votes
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
votes
1answer
134 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 ...
3
votes
1answer
60 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. ...
1
vote
1answer
50 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 ...
0
votes
1answer
65 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, ...
2
votes
1answer
36 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 ...
2
votes
1answer
48 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 ...
0
votes
0answers
36 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 ...
0
votes
1answer
54 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 ...
1
vote
1answer
57 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 ...
0
votes
1answer
20 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 ...
0
votes
1answer
52 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 ...
6
votes
2answers
208 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 ...
1
vote
1answer
41 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 ...
2
votes
1answer
103 views

A power series ring over $\mathbb C$

I have two questions around the ring of formal power series $R=\mathbb C[[x^2,x^3]]$. What is the global dimension of $R$? Is it a local regular ring? The global dimension of a ring is the ...
3
votes
1answer
47 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. ...
0
votes
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 ...
1
vote
1answer
29 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 ...
3
votes
3answers
140 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 ...
4
votes
3answers
86 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} ...
3
votes
1answer
43 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
votes
1answer
36 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 ...
0
votes
1answer
27 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 ...
0
votes
1answer
25 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, ...
2
votes
1answer
93 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). ...
0
votes
1answer
69 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 ...
1
vote
2answers
122 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 ...
3
votes
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 ...
6
votes
0answers
52 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 ...
1
vote
1answer
36 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 $S$ ...
1
vote
0answers
34 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 ...
3
votes
2answers
36 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 $I(J\cap K)=IJ\cap IK$?
2
votes
2answers
94 views

When does the regularity of $A$ implies the 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 ...
2
votes
1answer
41 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 ...
2
votes
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. ...
2
votes
1answer
83 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 ...
1
vote
1answer
35 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 ...
4
votes
1answer
101 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 ...
0
votes
1answer
29 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 ...
5
votes
1answer
99 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 $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
6
votes
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 = ...
2
votes
1answer
37 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 ...
6
votes
3answers
88 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 ...
1
vote
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
votes
1answer
44 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 ...