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

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46 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
2
votes
1answer
52 views

$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's Advanced Topics in The Arithmetic of Elliptic Curves (whose proof is left to the reader): Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
4
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0answers
54 views

Example of $A$-module but not $A$-algebra. [duplicate]

If $A$, $B$ are commutative rings, and if $B$ is an $A$-algebra then it is also an $A$-module. I am looking for an example that shows that the converse is not true. That is, I am looking for ...
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1answer
23 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be a $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
2
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0answers
35 views

Generalization of the Going up Theorem to arbitrary chains of prime ideals

Let $S$ and $R$ be commutative rings with $1$. This is the usual form of the Going up theorem that one encounters in commutative algebra texts: Let $S$ be integral over $R$, and suppose that we have ...
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1answer
50 views

Theorem 31.7 of Matsumura, Commutative Ring Theory

Theorem: If A is a Noetherian local ring and A[x] catenary, then A is formally catenary. In the proof, it is assumed that A is integral domain and A* (the completion of A) is not equidimensional and ...
8
votes
1answer
68 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
2
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0answers
43 views

When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
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0answers
43 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
3
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2answers
50 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
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1answer
39 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
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0answers
35 views

Prime ideal in indecomposable commutative ring [on hold]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
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0answers
48 views

Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...
3
votes
1answer
102 views
+50

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
1
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1answer
55 views

Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
2
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1answer
48 views

Galois group and traslations by rational numbers.

Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$. Now my question: ...
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0answers
26 views

When an intermediate ring $B$ is regular, where $A \subseteq B \subseteq C$ with $A$ and $C$ regular

Assume $A \subseteq B \subseteq C$ are noetherian integral domains. Further assume that $A$ and $C$ are regular rings (=noetherian ring such that every localization at a maximal ideal is a regular ...
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0answers
39 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 $R$ ...
1
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0answers
46 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 $$ ...
1
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1answer
80 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
votes
1answer
42 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
27 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 ...
3
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2answers
76 views

Semilocal commutative ring with two or three maximal ideals

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
4
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2answers
37 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 ...
0
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1answer
34 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 ...
1
vote
1answer
32 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 $R=S/I$. ...
1
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1answer
27 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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0answers
36 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 ...
0
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1answer
25 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
votes
1answer
44 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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2answers
45 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 ...
0
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0answers
39 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 ...
4
votes
2answers
67 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
votes
2answers
83 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 ...
0
votes
0answers
50 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 $x_ry_s\notin I$. ...
0
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1answer
25 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 ...
1
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0answers
55 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 ...
4
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0answers
47 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 ...
0
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1answer
30 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
33 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 ...
-1
votes
1answer
66 views

Example of flat module but not torsion free [closed]

I want an example of flat module but not torsion free. Does it exist? Please hint me. Thanks. Torsion submodule: if $R$ is a domain and $M$ is an $R$-module, then its torsion submodule is ...
-1
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0answers
27 views

An ideal with a special property [closed]

Please give me an example of an ideal $I$ of a commutative ring $R$ which has both of the following properties simultaneously: There are three non-unit elements $x_1, x_2, x_3$ of $R$ whose product ...
1
vote
1answer
38 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 $1$?
1
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1answer
30 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 ...
3
votes
2answers
263 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.
0
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1answer
19 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 ...
0
votes
1answer
27 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 ...
0
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0answers
54 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
41 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
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 ...