# Tagged Questions

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

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### Regular local ring and a prime ideal generated by a regular sequence up to radical

Let $R$ be a regular local ring of dimension $n$ and let $P$ be a height $i$ prime ideal of $R$, where $1< i\leq n-1$. Can we find elements $x_1,\dots,x_i$ such that $P$ is the only minimal prime ...
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### Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
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### Given a commutative ring $R$ and an epimorphism $R^m \to R^n$ is then $m \geq n$?

If $\varphi:R^{m}\to R^{n}$ is an epimorphism of free modules over a commutative ring, does it follow that $m \geq n$? This is obviously true for vector spaces over a field, but how would one show ...
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### Preimaging units to units

I'm interested in (unity-preserving) homomorphisms $f: S \to T$ between (commutative, with-unity) rings $S$ and $T$ so that if $f(x)$ is a unit, then $x$ was a unit to start with. For example, an ...
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### What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules "...
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### An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime (...
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### How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
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### Geometric meaning of completion and localization

Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions. The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
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### An inverse limit

Let $k$ be a field. Consider the inverse limit $\varprojlim k[x,y]/(y\cdot x^n)$. I wonder if there is a nice description of this ring? Geometrically, we look at the union of the line $y=0$ ...
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### When some polynomials in $\mathbb Z[X]$ determine a regular sequence in $\mathbb Z[X_1,\dots,X_n]$?

Let $f_1,\dots,f_n\in\mathbb Z[X]$ be non-constant polynomials (not necessarily distinct). Is it true that $f_1(X_1),\dots,f_n(X_n)$ is a regular sequence in $\mathbb Z[X_1,\dots,X_n]$? The trivial ...
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### Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
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### Characteristic of a finite ring with $34$ units

Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the Euler ...
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### Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
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### How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
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### How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?

Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can ...
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### If $R[x]$ and $R[[x]]$ are isomorphic, then are they isomorphic to $R$ as well? [duplicate]

There are examples of commutative rings $R \neq 0$ such that $R[x]$ is isomorphic to $R[[x]]$ (see this question; an example would be $R=S[x_1, x_2, \ldots][[y_1, y_2, \ldots]]$, with $S \neq 0$ any ...
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### Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

I'm trying to solve the following problem from Atiyah-Macdonald: Is the ring of continuous function on $[0,1]$ is Noetherian ? Certainly not, here are two non terminating ascending chain of ...
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### Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for effective ...
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### $B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
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### Every maximal ideal is principal. Is $R$ principal?

Let $R$ be a commutative ring with 1. If every maximal ideal of $R$ is principal, is $R$ a principal ideal ring?
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### Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
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Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid \... 2answers 470 views ### Tor and flat base change There is an interesting result in Rotman's homological algebra book. Let$A$be an$R$-module,$B$be an$(R,S)$-bimodule and$C$an$S$-module. (All rings commutative). Then Corollary 10.61 (in the ... 3answers 566 views ### A theorem due to Gelfand and Kolmogorov For any topological space$X$, we can define$C(X)$to be the commutative ring of continuous functions$f\,:\,X\rightarrow \mathbb{R}$under pointwise addition and multiplication. Then$C(-)$becomes ... 2answers 956 views ### Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13) I have been having trouble with an exercise in my abstract algebra course. It is as follows: Let$f: \mathbb{C}[x,y] \rightarrow \mathbb{C}[t]$be a homomorphism that is the identity on$\mathbb{C}...
When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$: Let $k$ be an algebrically ...