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
1
vote
1 answer

Primary ideals in completion $\hat{R}$

Let $(R,m)$ be a Noetherian local ring and $\hat{R}$ its $m$-adic completion. Then there is a one to one correspondence between $m$-primary ideals in $R$ and $\hat{m}$-primary ideals in $\hat{R}$. Suppose $I\in \hat{R}$ is an $\hat{m}$-primary…
Display name
  • 453
1
vote
1 answer

If $A$ is a $K$-algebra and a field, and is contained in an affine $K$-domain, then $A$ is algebraic over $K$

I am reading Lemma 1.1 in Kemper's book A Course in Commutative Algebra. I am stuck in understanding the proof of part (b) of this lemma. Some definitions in the books: By Affine $K$-algebra we mean a finitely generated algebra over a field $K$. By…
user628623
  • 39
  • 6
1
vote
0 answers

Prime characteristic $p$ and Power series ring

Let $R=K[[x_1,\ldots,x_n]]$ the power series ring over a field $K$ of prime characteristic $p$. If we take $K$ if $F$-finite, I know show that $R^{1/p}$ is a free $R$-module. If $K$ not is $F$-finite, is $R^{1/p}$ a free $R$-module? Notation.…
Wágner Badilla
  • 337
1
vote
1 answer

Noetherian graded ring is finitely generated $A_0$-algebra

Suppose $A=\bigoplus_{n=0}^\infty A_n$ is a Noetherian graded ring. $A_{+}=\bigoplus_{n>0}A_n$ is an ideal of $A$, hence finitely generated by homogeneous elements $x_1,...,x_s.$ Then, we want to prove that $A=A_0[x_1,...,x_s]$, and it is achived by…
julian
  • 123
1
vote
1 answer

Applying Nakayama's lemma is giving me trouble

Let $R$ be a noetherian commutative ring with unity, $I$ be a maximal ideal of $R$, and $R_I$ be the localization of the ring $R$ at $I$. I am having trouble understanding how the following follows from Nakayama's lemma (I'm using Atiyah…
mathfan24
  • 560
1
vote
1 answer

Prüfer domains are Arithmetical rings

Suppose $R$ be a Prüfer domain. How should I prove that it is an arithmetical ring?
sahra
  • 77
1
vote
0 answers

Generalising the definition of Primary Ideal

This isn't there in book. I am just curious. Let $R$ be a ring, then an ideal $I\subset R$ is called a primary ideal if $ab \in I$ implies either $a\in I$ or $b^n \in I$ for some $n\in \mathbb{N}$. Now I wanted to modify the definition. Let $R$ be a…
permutation_matrix
  • 1,356
1
vote
0 answers

Going-down property implies surjectivity

A ring homomorphism $f:A\to B$ is said to have going-down property if the going-down theorem hold for $f(A)$ as a subring of $B$. In the book Introduction to Commutative Algebra by Atiyah and Macdonald, there is an exercise to prove that if $f$ has…
julian
  • 123
1
vote
1 answer

"surjectively-free" from Matsumura, Commutative Algebra

One of my friend asked me about this exercise from Matsumura's "Commutative Algebra" (not "Commutative Ring Theory"), and I have no idea about the term "surjectively free" and how to solve this exercise. Let $A$ be a ring and $M$ an $A$-module. We…
Luka
  • 126
  • 6
1
vote
1 answer

Are flat local morphisms of local rings always faithfully flat injections?

I claim that flat local maps $(A, m_A) \to (B, m_B)$ are faithfully flat injections. To show faithful flatness of $B$ it suffices to show that $B \otimes_A k(m_A) = B/m_AB \neq 0$. However, by the local morphism definition, we know that $m_AB…
Daniel
  • 5,274
  • 1
  • 10
  • 25
1
vote
1 answer

Calculating the integral closure (normalizatrion) of a quotient over a polynomial ring in three variables.

Let $A=\mathbb{Q}[x,y,z]/((x+a)^2-z(y+z^b)^2)$ for some $a,b\in\mathbb{N}$. Assume that $A$ is an integral domain (this is easy to show, you can just use Generalised Eisenstein over $\mathbb{Q}[y,z][x+a]$ with prime ideal $(z)$ I think). The…
MarvinsSister
  • 81
  • 12
1
vote
1 answer

On the canonical morphism $B\otimes_A N \to N$?

Let $A$ and $B$ be two commutative rings with unit. Let $\rho: A \rightarrow B$ be a ring homomorphism and $N$ a $B$-module (and thereby a $A$-module). Question: Is there a canonical morphism $B\otimes_A N \to N$? If so, what is the expression…
Lelong Wang
  • 507
1
vote
1 answer

Cancellation of Projective Modules

Can you give me some example of finitely generated Projective A module P with rank 2 s.t. $P\oplus R[x] \cong P'\oplus R[x]$ and P is not isomorphic to P' where R is a local ring of dimension 3 and A=R[x]?
Dgarg12
  • 81
1
vote
0 answers

Explicit Example of Matlis Duality

In preparing for a talk, I am searching for an explicit example of Matlis duality. My attempt was to consider the Noetherian local ring $\mathbb{Z}_p$ which is the localization at the prime ideal $(p)$. Then take the injective hull of the residue…
Shrugs
  • 1,458
1
vote
1 answer

Prime ideal whose square is not primary

Let $R = k[x,y,z]$ and $P = (y^2-xz,x^2y-z^2,x^3-yz)$ ideal of $R$. Show that $P$ is prime ideal. Show that $P^2$ is not primary ideal. Hint: Show that $(x,y,z)\in \operatorname{Ass}_R(R/P^2)$
someone
  • 535
1 2 3
…
99 100