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
3
votes
2 answers

Nilpotent elements in commutative rings

Let $A$ be a commutative ring, $a, a+b \in A$ are nilpotent. Does this imply that $b$ is nilpotent?
Daniel Tum
  • 31
3
votes
0 answers

What does a minimal generator of an ideal mean?

For example in lemma 2.1 here Another example is here the phrase "a minimal generator" is used. I don't understand what this means in the absence of a specified set of generators. Can anyone explain
Pat
  • 31
3
votes
1 answer

Integral closure of 1-dimensional noetherian local domains

Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Clearly $S$ is $1$-dimensional noetherian semi-local domain. Is $mS=J(S)$, where $J(S)$ is the Jacobson radical of $S$?
alinaghi
  • 91
3
votes
1 answer

Both $R$ and $R/I$ are regular local rings

Let $R$ be a Noetherian local ring and $I$ is an ideal of $R$ such that both $R$ and $R/I$ are regular local rings. Could we deduce that $I$ is generated by an $R$-sequence? I know that a noetherian local ring is regular if and only if its maximal…
karparvar
  • 5,730
  • 1
  • 14
  • 36
3
votes
4 answers

In a Noetherian ring, does every set of generators of an ideal have a finite subset of generators

In a Noetherian ring, every ideal is finitely generated. Suppose an ideal $I$ in a Noetherian ring $R$ is generated by a set of generators $S$. If $S$ is infinite, does it have a finite subset that generates $I$.
Zuben
  • 183
3
votes
1 answer

Example of Noetherian ring over which the Euclidean algorithm is not valid.

As stated in the question, I am looking for a Noetherian ring over which the Euclidean algorithm is not valid. I am trying to construct non-trivial examples of Noetherian rings. Thank you.
algebraically_speaking
  • 67
3
votes
1 answer

R is a regular local ring of dimension $d$, and $I$ an ideal. If $R/I$ has depth $d − 1$, then $I$ is principal.

is this true? $R$ is a regular local ring of dimension $d$, and $I$ an ideal. If $R/I$ has depth $d − 1$, then $I$ is principal. if it is true please help me by a hint
user147308
3
votes
1 answer

Castelnuovo-Mumford regularity of a product

Let $k$ be a field, $R = k[x_1,\dots,x_n]$ the polynomial ring, $\mathfrak m = (x_1,\dots,x_n)$ and $M$ a finitely generated graded $R$-module. How can we see that $\operatorname{reg}(\mathfrak mM) \le \operatorname{reg}(M) + 1$, where…
Manos
  • 25,833
3
votes
0 answers

When does the inverse limit preserve the localisation?

Question When is the following true? $$\varprojlim(S_\alpha^{-1}A_\alpha)\cong(\varprojlim S_\alpha)^{-1}(\varprojlim A_\alpha)$$ (For details, consult the next part.) Notations One can consult this part only when needed. Let $I$ be a poset,…
awllower
  • 16,536
3
votes
0 answers

fundamental theorem of dimension theory for *local rings

Let $k$ be a field and $S=k[x_1,\dots,x_n]$ the polynomial ring with the usual grading. Let $M$ be a finitely generated graded $S$-module. Question 1: How can we see that $\dim M = 0$ implies that the length of $M$ is finite, i.e. $l(M)<…
Manos
  • 25,833
3
votes
2 answers

Ring of Convergent Power Series in R and C is a Local Ring

Let $k=\mathbb{C}$ or $\mathbb{R}$, and let k{x} denote the ring of power series with appropriate coefficients that are convergent around 0. Check that k{x} is a local ring. I have a similar…
math1234567
  • 1,736
3
votes
1 answer

Corollary 3.3.15 in Bruns and Herzog, Cohen-Macaulay Rings (self-contained question)

Theorem 3.3.14 [Bruns and Herzog, CMR]: Let $(R,m)$ be a CM local ring and $(R,m) \rightarrow (S,n)$ a flat local homomorphism. Then (a) If $\omega_R$ exists (this is the canonical module of $R$) and $S/mS$ is Gorenstein, then $\omega_S = \omega_R…
Manos
  • 25,833
3
votes
1 answer

Am I wrong in thinking it is isomorphism rather than homomorphism?

The following is a quotation from the proof of Proposition 11.10 in "Introduction to Commutative Algebra" by Atiyah and MacDonald. Also if ${\mathfrak m}'$ is the maximal ideal of $A'$, $A'/{\mathfrak m}'^n$ is a homomorphic image of $A/{\mathfrak…
Aki
  • 1,299
3
votes
1 answer

Wikipedia definition of an order (ring theory)

Wikipedia defines an order $\mathcal O$ of a finite type $\Bbb Q$-algebra $A$ to be a subring of $A$ satisfying the following properties. Here, by finite type $\Bbb Q$-algebra, I mean that $A=\Bbb Q[x_1,...,x_n]/I$. $\Bbb Q \mathcal O =…
Rodrigo
  • 7,646
3
votes
1 answer

What is meant by squeezing a module between two f.g. modules?

The author of the first answer in this thread of mathoverflow concluded that a module $K'$ was finitely generated because it was squeezed between two finitely generated modules. In and of itself, this statement, as I interpret, it is false, because…
Rodrigo
  • 7,646
1 2 3
…
99 100