# Tagged Questions

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

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### 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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### 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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### 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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### Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one: Are there any integrally closed BFDs which are not Krull domains? Some background information: A BFD (bounded factorization domain) is ...
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### 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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### 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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### 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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### 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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### 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 $r\in R$. ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### How do ideal quotients behave with respect to localization?

Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When ...
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### 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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### 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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### 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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### 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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### What's the “real” reason a finite map has finite fibers?

This is a soft question. I have encountered two very different proofs of what seems like "basically the same theorem," and I want to understand how they relate and "what the real explanation is." ...
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### 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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### 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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### Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
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### Determining the minimal number of generators of the maximal ideal of a local Noetherian ring

Let ($A,m$) be a local, Noetherian ring. If $n$ is the minimal number of generators of the unique maximal ideal $m$, then by Krull's Hauptidealsatz and Nakayama's Lemma, we have the following ...
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### Can every ideal have a minimal generating set?

Let $I$ be an ideal of commutative ring $A$ with unity. Does $I$ have a minimal generating set? At times, I am able to compute what they are for specific example, but it seems like it is true in ...
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### How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
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### 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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### 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 ...
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### Image of point of codimension one has codimension one?

I'm working on the following exercise (7.2.3) from Liu's Algebraic Geometry and Arithmetic Curves: Let $f: X \rightarrow Y$ be a morphism of Noetherian schemes. We suppose that either $f$ is flat ...
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### 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 ...