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

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Wanted: A purely algebraic proof of the frobenius theorem on distributions

Is there a purely algebraic proof of the frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of derivations of the algebra $R$ endowed with the lie ...
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0answers
26 views

Prime ideals in $A[x_1, \ldots,x_n]$

Let $A$ be a commutative, Noetherian ring and let us define a monomial ordering, $\prec$ on $A[x_1, \ldots,x_n]$. My doubt is regarding the maximal chain of prime ideals in $A[x_1, \ldots,x_n]$. When ...
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1answer
24 views

Show $R \setminus S$ is a union of prime ideals

I'm stuck on the following question: Let $R$ be a commutative ring with $1$, and $S \subseteq R$ a saturated multiplicative set (that is, $1 \in S$ and $x, y \in S$ if and only if $xy \in S$). ...
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1answer
29 views

Formal smoothness of $A \to A[t]/(h)$.

Let $A$ be a commutative noetherian ring, $T$ an indeterminate, $h=h(T) \in A[T]$, and $B:= A[T]/(h)$. When $B$ is formally smooth over $A$? (If $h$ is monic, is $B$ formally smooth over $A$?). ...
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21 views

A UFD ring for which the related formal power series ring is not a UFD

I know that the proposition $$ A \text{ is a UFD } \Rightarrow A[[X]] \text{ is a UFD }$$ is false. Wikipedia states that if $B=K[x,y,z]/(x^2+y^3+z^7)$ than $A=B_{(x,y,z)}$ is a counterexample but ...
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0answers
16 views

Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

Cross-posted from MO. At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons" though certainly there are others. ...
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0answers
30 views

When the fibers of $A \to A[T]/(h(T))$ are geometrically regular?

Let $A$ be an affine commutative noetherian domain over a characteristic zero field $K$, $T$ an indeterminate, $h=h(T) \in A[T]$ (not necessarily monic), $B=A[T]/(h)$, and assume that $(h)$ is a prime ...
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0answers
51 views

What do we call the ring homomorphism $R \rightarrow \mathrm{End}_{\mathbf{Ab}}(X)$ associated with an $R$-module $X$?

First, a convention: given an abelian group $X$, write $\mathrm{End}_{\mathbf{Ab}}(X)$ for the set of all group homomorphisms $$X \rightarrow X.$$ Now let $R$ denote a ring. Question. Given an ...
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1answer
34 views

Why do there is a unique continuous homomorphism?

Is this a right place to ask help for an exercise? Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...
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1answer
58 views

Need a counterexample: If the induced homomorphism $M/\mathfrak aM \to N/\mathfrak aN$ is surjective, then $f$ it's surjective.

This problem is from Atiyah and Macdonald, Introduction to Commutative Algebra, Exercise 10, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ ...
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1answer
24 views

Is a Noetherian normal local domain (universally) catenary?

Let $R$ be a ring. Then $R$ is $\textit{catenary}$ if for a pair of prime ideal $p \subseteq q$, all maximal chains of prime ideals $p = p_0 \subseteq p_1 \subseteq \dots \subseteq p_n = q$ have the ...
4
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1answer
85 views

Equivalent conditions for an ideal to be zero-dimensional.

For an ideal $I \subset \mathbb{C} [x_1, ... , x_n]$ show that dim$_{\mathbb{C}}R/I$ is finite iff $I$ is contained in only finitely many maximal ideals. Thoughts so far: I'm not sure how to get ...
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3answers
71 views

Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number.

I'm solving some exercises from my class notes on Commutative Algebra,On the following exercise I got stuck: Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number. As far as I ...
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2answers
123 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
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0answers
16 views

Definition of filtration over monoid

I want to know if the following definition is correct. A $\textbf{filtration}$ over a monoid $M$ (operation denoted multiplicativity), is a total order $<$ on $M$ that fulfills $1_M < x$ and if ...
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1answer
40 views

A question about finitely generated projective modules

Let $A$ be a commutative ring with unity and let $P$ be a finitely generated projective $A$-module. For $any$ $A$-module $M$, how does one show that $\operatorname{Hom}_A(P,A) \otimes_A M \simeq ...
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3answers
67 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
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0answers
26 views
+50

Which minimal hypothesis is necessary for a matrix on a fairly general ring to have a Jordan-Chevalley decomposition?

When I look at different proofs or expositions of the Jordan-Chevalley decomposition of a matrix, the minimal hypothesis I usually found is about the perfection of the field over which such ...
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1answer
54 views

Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay.

Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay. We have $\langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle=\langle x_1,x_3 \rangle \cap \langle ...
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0answers
29 views

Cohomological dimension of an arbitrary module.

In the paper, [P, Schenzel, On formal local cohomology and connectedness, J of Alg, 315 (2007), 894--923], he proves the following statement. (Corollary 2.2) Let $M$ be a finitely generated ...
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0answers
66 views
+50

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
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2answers
39 views

Formal power series over a regular ring is regular

I'm trying to prove that if $A$ is a regular ring then so is $A[[X]]$. The only proof I found of this statement is in Commutative Ring Theory by Matsumura, but it seems a bit over my knowledge so I'd ...
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1answer
38 views

An incorrect(?) proof of the Hilbert's Basis Theorem

This is my proof of the Hilber's Basis Theorem. I think it is incorrect. Because it is easier than other proofs. But I can't find out the mistake in my proof. Can anyone help me? Thanks! Claim If ...
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0answers
46 views

generalized affine scheme

I'm thinking about following theorem. For a finitaly algebraic theory $\mathbb{T}$, $\text{FP}\mathbb{T}$ denotes the full subcategory of $\mathbb{T}\text{-Alg(Set)}$ consisting of finitely presented ...
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0answers
28 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
4
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0answers
28 views

Which of the algebra isomorphisms hold?

Fix $m, n \ge 1$. Which of the algebra isomorphisms below hold? $k\langle t_1, \dots, t_m\rangle \otimes_k k\langle s_1, \dots, s_n\rangle \cong k\langle t_1, \dots, t_m, s_1, \dots, s_n\rangle$ $k[ ...
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1answer
20 views

Meaning of 3-disjoint

Definition: Two edges $\{x, y\}$ and $\{w, z\}$ of $G$ are said to be 3-disjoint if the induced subgraph of $G$ on $\{x, y, w, z\}$ consists of exactly two disjoint edges. (See page 5 of this file.) ...
4
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1answer
42 views

Converse of “localization at a prime is local”

Suppose $S^{-1}R$ is the localization of a ring R at a multiplicative subset S, and is local. Must S be the complement of a prime ideal?
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1answer
72 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
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0answers
19 views

The completion of the ring of Laurent polynomials with respect to the augmentation ideal.

Let $A = \langle a\rangle$ be an infinite cyclic group on one generator. I'm trying to understand the completion $\widehat{\mathbb{Q}}A$ of the group algebra $\mathbb{Q}A$ with respect to the ...
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1answer
57 views

Embedding tensor product of integral domains

Let $C$ be a subring of integral domains $A,B$ and let $C',A',B'$ denote their field of fractions respectively. Can we always embed $A\otimes_CB$ in $A'\otimes_{C'}B'$ by $a\otimes b\mapsto ...
1
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0answers
37 views

Primitive vectors in $A^n$

Let $A$ be a commutative ring with 1. Let n be a positive integer. I call a vector $(a_1,...,a_n) \in A^n$ primitive, if the ideal generated by $\{a_1,....,a_n\}$ is $A$. Question: Given a primitive ...
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0answers
59 views

Axiomatization of the equational theory of ideals in a commutative ring

Is there a known axiomatization of the equational theory of ideal operations in a commutative ring? I have in mind the following: Consider a language with operations for ideal intersection, product, ...
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1answer
125 views

Can an element in a Noetherian domain have arbitrarily long factorizations?

I tried to answer this question two days ago. Unfortunately, I said ring, rather than domain, which is what I wanted. So I try again. Let $R$ be a Noetherian commutative domain and let $r\in R$. ...
2
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1answer
33 views

Vanishing of Ext group and Krull dimension

Suppose $A=k[x_1,..,x_n]_{(x_1,..,x_n)}$, it is a regular local ring of dimension $n$. Let $B=A/I$ be a quotient ring of Krull dimension $r$. How to show $\operatorname{Ext}_A^i(B,A)=0$ for ...
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0answers
17 views

Extending McCoy's theorem to multiple indeterminates [duplicate]

So, working in a commutative ring with unity $R$, I've proven that $f\in R[x]$ is a zero divisor iff there exists $s\in R$ such that $sf=0$. I'm now being asked the followup question to extend ...
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1answer
18 views

Extending valuations and linear disjointness of fields

Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be ...
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0answers
15 views

Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
7
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1answer
150 views

Can an element in a Noetherian ring have arbitrarily long factorizations?

Suppose $R$ is a Noetherian ring. Is it possible that an element $r\in R$ have arbitrarily long factorizations? That is, for all $n>0$, is there a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ such ...
2
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1answer
38 views

Discrete convolution of two sequences

Let $R$ be a commutative ring with unity. A finite sequence $x=\left< x_0,\dots,x_n\right>$ with elements in $R$ is called to be prime if there exists $a_0,\dots,a_n \in R$ such that ...
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0answers
27 views

Can the torsion of a formal group have rank going to infinity?

Suppose $F(x,y) \in \mathbb{Z}_p[[x,y]]$ is a formal group over $\mathbb{Z}_p$. I denote with $G(-)$ the corresponding functor, so that $G(K)$ will denote for me the maximal ideal of $O_K$ equipped ...
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2answers
89 views

Are these quotient modules isomorphic?

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder ...
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1answer
66 views

Maximal ideals and the projective Nullstellensatz

This is a simple question, but it's one of those things that I've been thinking about so much that I've just kind of lost where I am and need some explicit reference. One of the main corollaries of ...
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0answers
43 views

Question about points of a variety lying in an extension as K-morphisms

I hope that someone can shed some light on this for me.. or at least point me to some references. Suppose that $X$ is an algebraic (let's just say affine) variety defined over $k$. Suppose I have a ...
4
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1answer
59 views

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$.

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$. Here's what I tried: If $1\otimes (1,1,\ldots)= 0$, then $1\otimes ...
2
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2answers
62 views

What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?

Suppose $B$ and $C$ are commutative rings, $A$ a $B$-algebra, and $B$ is a $C$-module. What exactly is the "adjunction" isomorphism $$Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))?$$ Given $A\to C$, it needs to ...
0
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1answer
71 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
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1answer
30 views

Gauss lemma for arbitrary commutative ring [duplicate]

Part (iv) of exercise #2 for chapter 1 in Atiyah and Macdonald's book Introduction to Commutative Algebra asserts that if $f, g \in A[x]$ are primitive then $fg$ is primitive. We know that this is ...
3
votes
1answer
53 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
3
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1answer
24 views

Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.

Let $R$ be a local commutative ring with the maximal ideal $M$. Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$. I tried to apply ...