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

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44 views

An integrally closed subdomain of a polynomial ring

Let $\mathbb{C} \subset R \subset \mathbb{C}[x,y]$ be a noetherian integral domain. Further assume that: (1) $\mathbb{C}[x,y]$ is separable over $R$. (2) $\mathbb{C}[x,y]$ is algebraic over $R$ ...
2
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0answers
50 views

Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ ...
2
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1answer
101 views

$R$ is normal. Are $R[x]$ and $R[[x]]$ normal?

Studying about normalizations I've bumped in the following theorem: Theorem. Let $R$ be a normal (integrally closed) domain, then $R[x]$ is a normal domain. How to prove (elegantly, if possible) ...
2
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1answer
44 views

Flatness of closure of subring

Assume we are given Noetherian local rings $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ such that: $A \subset B$ and $\mathfrak{m} = A \cap \mathfrak{n}$, $B$ is a finite $A$-module. It is known ...
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1answer
30 views

Primary decomposition of modules - uniqueness proof

Let $M$ be $A$-module, $A$ commutative ring, and $N$ submodule and let $$N=Q_1\cap\dots\cap Q_r=Q'_1\cap \dots \cap Q'_s$$ be reduced primary decompositions of $N$. Then $r=s$. The set of primes ...
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2answers
92 views

Semilocal commutative ring with two or three maximal ideals [closed]

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
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2answers
41 views

Can a the variety associated to a finitely generated $K$-subalgebra of $K[X]$ be embedded into $\mathbb{A}^3$?

Let $K$ be a field. Is there an example of a finitely generated $K$-subalgebra $$ A\subseteq K[X] $$ which is not isomorphic to $K[T_1,T_2,T_3]/I$ for some ideal $I$? As $A$ is finitely ...
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1answer
38 views

A question about the module of differentials [closed]

I want solve this good exercise: Let $(S,m)$ be a regular local ring that is the localization at a maximal ideal of a finitely generated algebra over a field $k$, and let $x_1, \ldots, x_d$ be a ...
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1answer
38 views

A regular sequence in a determinantal ring

Let $S=K[X_{ij}\colon 1\le i\le m, 1\le j \le n, m\le n]$ be a ring of polynomial with coefficient in a field, $X=(X_{ij})$ a matrix of indeterminates, $I$ the ideal of maximal minors and $R=S/I$. ...
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1answer
31 views

Is the dimension of a finitely generated $K$-subalgebra of $K[X_1,\ldots,X_n]$ bounded above by $n$?

Let $K$ be a field. Is there an example of a finitely generated $K$-subalgebra $$ A\subseteq K[X_1,\ldots, X_n] $$ of Krull dimension $\dim A>n$? If yes, is there such an example for $n=1$?
3
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1answer
65 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
2
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1answer
38 views

Projective dimension of monomial ideal

Definition. The support of a monomial $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ is defined to be the set $\operatorname{supp}(x_1^{\alpha_1}\cdots x_n^{\alpha_n})=\{x_i: \alpha_i >0\}$. Let ...
2
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1answer
46 views

Calculating the coordinate ring and irreducible components

Consider the graded ring $S=(R/I)\oplus (I/I^2)\oplus (I^2/I^3)\oplus\cdots$ Take $R=k[X,Y],I=(X^2Y,XY^2)$. Then $S=k[X,Y]/(X^2Y,XY^2)\oplus(X^2Y,XY^2)/(X^2Y,XY^2)^2\oplus\cdots$. I am not sure ...
2
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1answer
53 views

an example of a module that is not injective

I know that since $\mathbb Z$ is a PID hence every free module is projective and conversely. Hence since $\mathbb Q$ is not free as a $\mathbb Z-$ module then it is not projective. But is $\mathbb ...
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0answers
41 views

How bad must be a ring to allow cyclic artinian modules that are not noetherian?

I've been studying the relations between artinian and noetherian modules over commutative rings. One can prove two interesting results for the commutative case. Theorem Every commutative artinian ...
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2answers
74 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
4
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2answers
89 views

Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module?

Let $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$ be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that ...
1
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1answer
66 views

$R/I$ satisfies $S_2$ conditions

Let $R=k[x_1,...,x_{n},y_1,...,y_n]$ be a ring over $k$ and $I=\langle \{x_iy_j|$ for some $i,j \in\{1,...,n\}\}\rangle$ be ideal of $R$ and there are $r,s\in\{1,...,n\}$ such that $x_ry_s\notin I$. ...
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1answer
26 views

Inverting a nonzerodivisor of a module

I'm reading the Paper "What makes a complex exact?" by Eisenbud and Buchsbaum. On page 266 it says: Thus we may assume $0 \neq \operatorname{rank}(\phi_n,L) < \operatorname{rank}(F_n)$ and ...
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0answers
58 views

Quotient field - base change

For my master thesis, I need to examine the following statement: $Frac(R) \otimes_{k} L \cong Frac(R \otimes_{k} L)$, where $R$ is an integral domain over the perfect field $k$ and $L$ is a finite ...
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0answers
49 views

Rank of tensor product of morphisms

Let $R$ be a commutative, noetherian, unital ring, $F$ and $G$ two projective $R$ modules, $\phi: F\to G$ a module morphism and $M$ a finitely generated $R$ module such that $$\phi \otimes M := \phi ...
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1answer
33 views

Radical ideal in $\mathbb{R}[x,y,z]$

In $\mathbb{R}[x,y,z]$ is the ideal $I=\left\langle xz,yz\right\rangle$ radical? If $f \in I$ tried write $f=g.xz+h.yz+ax+by+c$ and conclude that $f^m \notin I$, if $m>0$, but I could not.
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0answers
35 views

Connection between local freeness and the rank of matrices

I am reading ch.16 of Eisenbud's Commutative Algebra, more precisely it's the very first paragraph of 16.7, where he wants to prove: Suppose that $\mathcal{J}: R^t \longrightarrow R^r$ is a map of ...
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1answer
42 views

How does one find the Krull dimension of a composite ring?

For example, if the ring is $\mathbb{Z} + X \mathbb{Q}[X]$. Is the dimension $1$?
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1answer
31 views

Finitely generated projective modules over a simple algebraic ring extension of a polynomial ring

The well-known theorem of Quillen-Suslin says that a finitely generated projective module over $k[x_1,\ldots,x_n]$ is free, See ...
4
votes
2answers
275 views

Example: Krull dimension 1 but not a PID

It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.
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1answer
20 views

Example of noetherian module non-uniquely expressible as sum of indecomposable submodules

If $M$ is a noetherian module then it can be written as a finite sum of indecomposable submodules of $M$. The same can be concluded if we assume instead $M$ to be artinian. If we ask for both $M$ to ...
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1answer
29 views

One dimensional integral domains are Cohen-Macaulay

A $1$-dimensional integral domain is always Cohen-Macaulay (C-M). I know this fact, but I do not know how can I reach at. Maybe one should use, somehow, the fact that $R$ is C-M if and only if each ...
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0answers
58 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
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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, ...
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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
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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
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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
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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
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
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1answer
54 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
48 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
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
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1answer
35 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
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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 ...
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0answers
35 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
52 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
55 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
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
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1answer
51 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
207 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
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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
102 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
46 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
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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 ...