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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
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disputing a length equality in Matsumura (fundamental theorem of dimension theory)

Let $A$ be a semilocal Noetherian ring with Jacobson radical $m$ and $M$ a finite $A$-module. Let $x \in m$. According to Matsumura's Commutative Ring Theory p. 99 (Step 2), $l(xM/xM\cap m^n M)=l(M/(m^nM:x))$. It seems to me that this equality is…
Manos
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Flat algebras over Noetherian local rings

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. Is there a commutative flat $R$-algebra $S$ such that $\mathfrak{m}S=0$ or $S/\mathfrak{m}S$ has finite flat dimension over $S$? I know that if $R$ is regular of prime characteristic,…
CARLO
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Upper bound for arithmetic rank of ideals in a local domain

Does anyone knows how to prove or disprove that if $(R,m)$ is a Noetherian local domain of dimension $d$ one has that $\mu(I)\leq d$ for every ideal $I\subset m$? Where the arithmetic rank of $I$ is $\mu(I)$ and $\subset$ means strictly contained.
yo yo
  • 536
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An infinite dimensional local ring with finite residue field

It is well-known that there is examples of infinite (Krull) dimensional ring. For example, Kang and Park, in [Example, pages 111 and 112, A localization of a power series ring over a valuation domain , JPAA 140 (1999) 107-124], constructs an…
T. Ali
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A conventional problem of regarding $k$ as an $k$-algebra

When I was reading this post, it was mentioned that a field $k$ is an initial object in the category of $k$-algebras. But if I understand things correctly, this seems to rely on some (rather reasonable) convention. A $k$-algebra is simply a ring…
Ray
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Question about inverse limits.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On Page 104, I have some questions about the proof that $\{A_n\}$ is surjective implies $d^A$ is surjective. We have to prove that given $(a_n) \in A = \varprojlim…
LJR
  • 14,520
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Proving that $f_1(1)=1$ for a multiplicative $R$-linear morphism $f_1: A \to B$

This question regards the proof of proposition 8 in chapter 8.1 of Bosch's 'Algebraic Geometry and Commutative Algebra'. Let $A, B$ be $R$-algebras, $f_0: A \to B$ a morphism of $R$-algebras (that is, an $R$-linear homomorphism of rings), and…
user480840
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Quotient by powers of a principal maximal ideal is Artinian

Let $R$ be a commutative ring (if necessary we assume it is an integral domain), and $\mathfrak{m}=(f)$ be a maximal ideal that is principal. Is it true that $R/\mathfrak{m}^n$ is a local Artinian for all $n>0$? I can see that it is local since…
CO2
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Questions about Grothendieck groups.

I have a question of the exercise 26 on page 88 of the book introduction to commutative algebra by Atiyah and Macdonald. In 26(iii), let $A$ be a field. Then finitely generated $A$-modules are finite dimensional $A$-vector spaces. Two finitely…
LJR
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Good presentation of the Witt numbers

Let $A$ be a $\mathbb{F}_p$-algebra of finite presentation. Then we know that $W(A)$ is a $W(\mathbb{F}_p)=\mathbb{Z}_p$-algebra. My question is if there is a "good" presentation of $W(A)$ as $\mathbb{Z}_p$-algebra. If it is helpful for this, I'm…
curious math guy
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Proposition 1.6 Atiyah

Let $A$ be a ring and $\mathcal{m}$ a maximal ideal of $A$, such that every element of $1+\mathcal{m}$ is a unit in $A$. Then $A$ is a local ring. Let $x\in A\setminus\mathcal{m}$ (It is not the quotient ring, but $A$ setminus $\mathcal{m}$).…
user805324
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Questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.

I have some questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald. On Line 8-9 of Page 42, it is said that $(xs-a)t=0$ for some $t\in S$ iff $xst\in \mathfrak{a}$. If $(xs-a)t=0$, then $xst=at \in…
LJR
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Question about the book introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.

On Line 2 of Page 40 of the book introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald, it is said that $m/s =0$ implies $tm=0$ for some $t \in S$. I think that if $m/s=0$, then $m/s = 0/t_1$ for some $t_1 \neq 0, t_1 \in S$.…
LJR
  • 14,520
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Why is $k \rightarrow A \rightarrow A / I$ and isomorphism of rings if $I \subset A$ is maximal?

Let $k$ be a algebraically closed field, $A$ a finitely generated $k$-Algebra and $I \subset A$ a maximal ideal. Let $\varphi: k \rightarrow A$ be a ring homomorphism. Why is this combination $$k \rightarrow A \rightarrow A / I$$ an isomorphism?
Haderlump
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Prime ideals and Zorn's lemma

Let $A$ be a ring, $x$ a nonzero element of $A$ and consider the annihilator of $x$, i.e $Ann(x)$. Now let $S$ denote the collection of all prime ideals of $A$ containing $Ann(x)$. It can be shown using Zorn's lemma that $S$ contains minimal…
user6495
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