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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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Strategy to prove that rings are UFD's

I was looking for strategies to prove when rings are going or not to be UFD's. I really only know that if I manage to prove that there is an element on the fraction field $K$ of my ring $R$ that is not integral (meaning that it does not solve a…
Daniel Montealegre
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Is this quotient of a polynomial ring local?

Let $k$ be a field, $R = k[X_1, \ldots, X_n]$ and $I = \langle X_1, \ldots, X_n\rangle$. I am trying to prove that the quotient ring $R/I^r$, $r\in\mathbb{N}$ is local, i.e., has only one maximal ideal. Here we define $I^r$ as the ideal generated by…
Eric Vaz
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Does every closed set of prime ideals of a noetherian ring contain a finite dense subset?

Does every closed set of prime ideals of a noetherian (commutative) ring contain a finite dense subset? EDIT 1. Here is the motivation. In books like Mumford’s red book or Eisenbud-Harris, the authors describe the topology of the spectrum of some…
Pierre-Yves Gaillard
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Is there a proof of the Krull Intersection Theorem without using the Artin-Rees lemma?

Textbooks on commutative algebra always prove the Krull Intersection Theorem using the Artin-Rees lemma. But is it really necessary to do so? Can one prove the Krull Intersection Theorem without using the Artin-Rees lemma? Is it even possible? I…
user107952
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Can a ring have both finite and infinite residue field

Let $R$ be a commutative ring with unity. Assume $R$ is integral (i.e. it has no zerodivisors, i.e. it is reduced and $\operatorname{Spec} R$ is irreducible) and noetherian. Let $\mathfrak{m}, \mathfrak{n} \subseteq R$ be two maximal ideals. Can it…
pmp
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Determine whether two rings are isomorphic

$k$ is a field, char$(k)\neq 2$. Can $k[x,y]/(y-x^2)$ be isomorphic to $k[x,y]/(x^2+y^2-1)$ by some ring homomorphism? Edit: Since jspecter gives a nice and quick proof, I will ask another question Can $k[x,y]/(y^2-x^2(x+1))$ be isomorphic to…
wxu
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Krull dimension of a ring which is finitely generated module over a finitely generated $k$-algebra

Let $k$ be a field and $R=k[y_1,\cdots,y_d]$ where $y_i$ are algebraically independent over $k$. Suppose that $S=k[x_1,\cdots,x_d]$ is a subring of $R$ such that $R$ is a finitely generated $S$-module. It is well known that the Krull dimension of…
Manos
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Finitely generated $k$-modules have finite spectrum

I've been doing an exercise, and reduced the problem to the following claim. I don't know how to prove it yet, hints are appreciated. Let $R$ be a ring and $k$ be a field such that $R$ is finitely generated as a $k$-module, meaning there exists an…
Zach Hunter
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a question on the minimal prime divisors of an ideal

This question is motivated by the second part of Step 1 in the proof of Theorem 14.14 in Matsumura's Commutative Ring Theory, p. 112. Let $k$ be an infinite field and $Q$ a homogeneous ideal of $k[x]=k[x_1,\cdots,x_s]$. Suppose that…
Manos
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Question about Hilbert-Samuel function (Commutative Algebra prop.12.2)

I'am reading Eisenbud, Commutative Algebra, p.272, Proposition 12.2 Why can we assume that $R$ is graded? Here, parameter ideal is defined in his book p.235. Or, in other version of his book, he call 'ideal of finite colength', instead of…
Plantation
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Atiyah-Macdonald Proposition 1.2

Proposition 1.2. Let $A$ be a ring $\neq 0$. Then the following are equivalent: i) $A$ is a field ii) the only ideals in $A$ are $0$ and $(1)$ iii) every homomorphism of $A$ into a non-zero ring $B$ is injective. I am confused by how ii) implies…
Milan
  • 133
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degree of the Hilbert polynomial of a quotient

Let $A=\bigoplus_{n \ge0} A_n$ be a Noetherian graded ring with $A_0$ Artinian. Suppose that $A=A_0[a_1,\dotsc,a_d]$ with $a_i$ having degree $1$. Let $M$ be a finitely-generated graded $A$-module. Then its Hilbert polynomial $\phi(n)_M$ is defined…
Manos
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Does product distribute with respect to intersection for ideals in a ring.

Let $I,\, J$ and $K$ be ideals in a commutative ring $R$. Could you please give an example such that $(I\cap J)K = IK\cap JK$ is not true?
PK Sharma
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Prime ideal in the ring of polynomials

I'm trying to do the following: Let $R = K[X,Y,Z]$ and $\mathfrak{p}$ = $(X+Y,Z^{2}-X)$. Show that $\mathfrak{p}$ is prime and find the transcendence degree of $R/\mathfrak{p}$. If I prove that $\mathfrak{p}$ is prime the question is over just…
User43029
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Geometric meaning of $\operatorname{Spec}(k[x_1,\ldots, x_n])$

I am a beginning learner of commutative algebra, using the book commutative algebra by Matsumura. In the book he often refers $\operatorname{Spec}(k[x_1,\ldots, x_m])$ to an affine plane where $k$ is a field. But I do not understand how this…
JKDASF
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