Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [commutative-algebra]

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

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
9
votes
3 answers

Integral closure of $\mathbb{Q}[X]$ in $\mathbb{Q}(X)[Y]$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of $\mathbb{Q}(X)$ given by $K:=\mathbb{Q}(X)[Y]$, where…
Federica Maggioni
  • 8,422
9
votes
1 answer

Atiyah-MacDonald exercise 3.29

I have a question regarding the exercise 3.29 in Atiyah-MacDonald. In the previous exercises it was introduced the constructible topology. Basically the space $\DeclareMathOperator{\Spec}{Spec}\Spec(A)$ for a ring $A$ is endowed with a topology such…
Sniper
  • 168
9
votes
2 answers

Dimension of the total ring of fractions of a reduced ring.

Let $A$ be a commutative reduced ring (need not be noetherian). Let $S$ be the set of all non-zerodivisors of $A$. What is the Krull dimension of $S^{-1}A$ ?
manoj
  • 233
9
votes
1 answer

$K[[x]]$ is not a Jacobson ring

Recall that a ring is called Jacobson if the radical of an ideal in the intersection of the maximal ideals that contains it (this is always true with prime ideals). $K[[x]]$ is not Jacobson. I know that this ring is local with $(x)$ as its only…
user56741
  • 141
9
votes
2 answers

Finite number of elements generating the unit ideal of a commutative ring

Let $A$ be a commutative ring with $1$. Let $f_1,\dots,f_r$ be elements of $A$. Suppose $A = (f_1,\dots,f_r)$. Let $n > 1$ be an integer. Can we prove that $A = (f_1^n,\dots,f_r^n)$ without using axiom of choice? EDIT It is easy to prove $A =…
Makoto Kato
  • 42,602
9
votes
2 answers

Is Nullstellensatz true for arbitrary fields if there aren't hidden points?

The ideals $I=(X,Y)$ and $J=(X^2+Y^2)$ in $\mathbb R[X,Y]$ are such that $V(I)=V(J)$ and their radicals aren't the same contradicting the Nullstellensatz (in case it was true for arbitrary fields). However, this shouldn't be a surprise, if we look…
Gabriel Furstenheim
  • 794
9
votes
2 answers

Automorphism of a formal power series ring

Is the following theorem true? If yes, how would you prove it? Theorem Let $A$ be a commutative ring. Let $A[[x]]$ be the ring of formal power series in one variable. Let $\mathfrak{m}$ be the ideal of $A[[x]]$ generated by $x$. Let $u$ be an…
Makoto Kato
  • 42,602
9
votes
2 answers

Why is a strict $p$-ring whose residue ring is a field necessarily local?

Let $A$ be a strict $p$-ring. Recall that this means $A$ is $p$-adically separated and complete, $p:A\rightarrow A$ is injective, and $A/pA$ is a perfect $\mathbf{F}_p$-algebra. If $A/pA$ is a field, then $A$ is known to be a discrete valuation…
Keenan Kidwell
  • 26,020
9
votes
1 answer

Necessary and sufficient condition that a localization of an integral domain is integrally closed

Is the following proposition true? Proposition? Let $A$ be an integral domain, $K$ its the field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. Let $I = \{a \in A; aB \subset A\}$. Let $P$…
Makoto Kato
  • 42,602
8
votes
2 answers

Invertible modules are locally free of rank 1

The original context of the following question is something about coherent sheaves over noetherian schemes, but the question itself is purely (commutatively-)algebraic. The definition of an invertible module is usually taken to be "locally free of…
KotelKanim
  • 2,856
8
votes
1 answer

Localization at finitely many minimal prime ideals

Let $A$ be a commutative ring with finitely many minimal prime ideals $\{p_1,\dots,p_n\}$. Let $A_{p_1,\dots,p_n}$ be the localization of $A$ away from the minimal primes, i.e. $S^{-1}A$ where $S = A-\bigcup_{i=1}^np_i$. Question 1 (Answered): A…
Alex Kruckman
  • 76,357
8
votes
1 answer

Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5)

This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows: Let $A=k[X,Y]/(Y^2-X^2-X^3)$. Prove that the…
Numbersandsoon
  • 2,910
8
votes
1 answer

"Instructive" proof of "If I is maximal among ideals not ..., then I is prime"

In this question all rings are commutative with identity. Consider the following well-known statement: (*) Let $R$ be a ring and $S$ a multiplicatively closed subset of $R$. Suppose $I$ is an ideal of $R$ maximal among those not meeting $S$. Then…
Tom Bachmann
  • 1,146
8
votes
1 answer

Question about Zariski topology

Here is the question: Let $A$ be a commutative ring with unit, $X=\mathrm{Spec}A$, $U_i$s be quasi-compact open sets of $X$ such that $\emptyset=\cap_{i\in I}U_i$, then there is a finite subset $I_0$ of $I$ such that $\emptyset=\cap_{i\in…
wxu
  • 6,671
8
votes
1 answer

Question about a proof on Atiyah Macdonald

I have a question about a step of a proof in Atiyah Macdonald. It's the proposition 2.4. Let M be a finitely generate A-module, let a be an ideal of A, and let $ \phi $ be an A-module endomorphism of M such that $$ \phi \left( M \right) \subset…
Daniel
  • 3,053
1 2 3
…
99 100