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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.

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Maps of maximal ideals

Prove that $\mu:k^n\rightarrow \text{maximal ideal}\in k[x_1,\ldots,x_n]$ by $$(a_1,\ldots,a_n)\rightarrow (x_1-a_1,\ldots,x_n-a_n)$$ is an injection, and given an example of a field $k$ for which $\mu$ is not a surjection. The first part is clear,…
Bombyx mori
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Intersection of infinitely many prime ideals

I was thinking of the fact that, if you take an infinite number of prime ideals $p_i\mathbb{Z}$ in the ring $\mathbb{Z}$, then their intersection is zero. What kind of commutative rings satisfy that the intersection of an infinite number of prime…
Marco Flores
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What is a generic element in a ring

I am now reading a commutative algebra paper, in which the name "generic element" of a commutative ring appears, however, I can not find the definition in that paper, and also my commutative algebra textbook. So, could you please tell me what is a…
knot
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Ex. 2.6 from Eisenbud: localization and infinite intersections

Can someone please explain to me why, given an infinite field $k$, we have that $$\left(\bigcap_{a \in k}{(x-a)}\right) [U^{-1} ] \neq \bigcap_{a \in k}{\left((x-a)[U^{-1}] \right)},$$ where $U$ is the set of nonzero elements of $k[x]$?
Hadji
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Proving elements of a polynomial ring are integral over another.

I have a quotient polynomial ring $ R = k[X,Y,Z]/ \langle X^2 - Y^3-1, XZ-1 \rangle$ where $k$ is a field and $X,Y,Z$ are variables. Let $x, y, z $ be the images of $X,Y,Z$ respectively. Fixing $a, b \in k$ and writing $ t = x +ay +bz$, I need to…
Vishesh
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Finding integral closure of $k[x]$ in $k(x)[y]/(y^2-x^3+x)$

There is a similar problem here at: Find the integral closure of an integral domain in its field of fractions The problem from the link has a nice proof because it has a parametrization of $t=\frac{y}{x}$, and it's easy to show the integral closure…
ZTransformer
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Noetherian hypothesis when permuting elements of a regular sequence?

One of the first results showed when studying regular sequences is that we are allowed to shuffle the elements of the sequence if the ring is noetherian, local, and the module is finite (see Proposition 2 in here for a proof). I know the ring being…
user347489
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Imbedding of a reduced ring into a direct sum - Why reduced?

This pertains to Ex. 1.13 (self-studier) in Reid's "Undergrad. Commutative Algebra": If $A$ is a reduced ring and has finitely many minimal prime ideals $P_i$ then $A\hookrightarrow \bigoplus_{i=1}^n A/P_i$; moreover, the image has nonzero…
user12802
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Is the associated graded ring of a normal domain a domain?

Let $R$ be a Noetherian local ring, $\mathrm{gr}(R)$ be its associated graded ring. I know that when $R$ is regular, then $\mathrm{gr}(R)$ is a domain (in fact a polynomial ring). I also know an example where $R$ is a domain but $\mathrm{gr}(R)$ is…
Yifeng Huang
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Is it true that $\sqrt{I}/I$ is a finite-dimensional vector space?

Let $R$ be the polynomial ring in $n$ variables over an algebraically closed field $k$. I'm trying to prove that for all ideals $I$ of $R$ it holds that $R/I$ is a finite-dimensional $k$-vector space of and only if $R/\sqrt{I}$ is as well. I need…
Daniel
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Rank k submodules of $\mathbb{Z}^n$

Let $M$ be a rank $k$ submodule of $\mathbb{Z}^n$. When is its image in $\mathbb{F}_p^n$ a rank $k$ subspace (where $p$ is any prime)? Can this situation be characterized in terms of a flatness condition?
Madeline Brandt
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Potential alternative definition of discrete valuation ring

Douady and Douady's "Algèbre et théories galoisiennes" proposes an exercise where we must show that different definitions of a discrete valuation ring are equivalent (3.1.10.b): Let $A$ be a ring and $w : A \to \mathbb{N} \cup \{\infty\}$ be a…
Arthur Azevedo De Amorim
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Proposition 7.8 in Atiyah MacDonald

I am reading "Introduction to Commutative Algebra" by Atiyah and MacDonald, and I am stuck at a detail in Proposition 7.8: Let $A\subset B\subset C$ be rings. Suppose that $A$ is Noetherian, that $C$ is finitely generated as $A$-algebra and that $C$…
user427787
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Castelnuovo-Mumford regularity and filter regular sequences

Well, I have asked this question on MO, I do not like asking duplicate question on both sites, however, my question might be too elementary for them, so I decided to post it here. Sorry if you feel it is not appropriate. Let $A$ be a commutative…
knot
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A doubt in the proof of Noether's normalization lemma.

My question comes out of Exercise 5.16 of the book by Atiyah-Macdonald. Let $k$ be an infinite field and let $A \neq 0$ be a finitely generated $k$-algebra. Let $x_1, \dots , x_n$ generate $A$ as a $k$-algebra. We can renumber the $x_i$ so that…
Jeong
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