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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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What would be the rank of the ring $R$ over $S$?

Let $\mathbb{F}_p$ be the finite field of characteristic $p>0$. Consider the ring of power series $R=\mathbb{F}_p[[x_1,x_2, \cdots, x_n]]$ and its free subring $S=\mathbb{F}_p[[x_1^{p^{i_1}}, x_2^{p^{i_2}}, \cdots, x_n^{p^{i_n}}]]$ for some positive…
MAS
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Maximal ideals of $\mathbb F_2+T\mathbb F_4[[T]]$

I try to determine all the maximal ideals of the domain $\mathbb F_2+T\mathbb F_4[[T]]$. I even do not know whether it is a Dedekind domain. In case it is not, can one determine its prime ideals too? Thanks in advance for any hints or solutions.
joaopa
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Example of a flat module whose module of fractions is not flat

Is it true that if $A$ is a commutative ring, then the module of fractions $S^{-1}M$ of a flat $A$-module $M$ is a flat $S^{-1}A$-module? This is certainly true for localizations at primes, but I'm unsure if this holds in general. I suspect that…
mathyokan
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Corollary of 1.11 in Atiyah and Macdonald

I would like to prove the following variation of 1.11 in AM. If $\mathfrak{p_1},\ldots,\mathfrak{p_n}$ are prime ideals and $\mathfrak{a}$ an ideal such that $\mathfrak{a}\neq\mathfrak{p_i}$ for each $i$. Then $\mathfrak{a}\neq\bigcup…
coolpenguin
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Kahler differentials of quotient field of regular local ring

Let $A$ be a regular local ring, $k$ its residue field. Assume $k$ is perfect and $A$ is the localization of finitely generated algebra. Then $\Omega_{A/k}\otimes_A K \cong \Omega_{K/k}$. I want to show this to be true. I can see how the result…
coolpenguin
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Module of Kahler differentials of an affine variety

Here's a question I am working on but I'm unable to make sense of it because the notation is used very liberally. This question is based out of the chapter about Derivations in Matsumura, Commutative ring theory. If $X$ is a variety over a field $k$…
coolpenguin
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Dummit & Foote 15.4 prop 46(5)

Let $R$ be commutative with 1. If $M\subset R$ is maximal, $I$ is $M$-primary, then $R_M/^eI = R/I$. In particular, $R_M/^eM = R/M$ and $^eM/(^eM)^n=M/M^n$. I am stuck in the last statement. So the second statement follows from the first directly.…
Jun Xu
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Relation between transcendental dimension and Kähler Differentials

I realise this is a strange question but my supervisor hasn't been replying to my emails. I am trying to understand the relationship between the transcendental dimension of irreducible affine varieties (as in Atiyah and MacDonald, Chapter 11) and…
coolpenguin
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Support of a non-trivial local ring is the whole spectrum?

$\newcommand{\Supp}{\operatorname{Supp}}\newcommand{\Spec}{\operatorname{Spec}}$Suppose $A$ is a local ring. There are a few places where $\Supp(A)=\Spec(A)$ seems necessary, for example $V(x)= V(x)\cap \Spec(A)=V(x)\cap \Supp(A)=\Supp(A/xA)$, also…
Jun Xu
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Isomorphism between localisation

Let $R$ be a unital commutative noetherian ring. Let $x_1,x_2$ be two non zero divisors of $R$. Is it real that $\frac{R_{x_1x_2}}{R_{x_1}+R_{x_2}}\cong_{_{R-mod}}\frac{R_{x_1}}{R}\otimes_R\frac{R_{x_2}}{R}$? The problem is that i have some…
yo yo
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Can i take $-1$ instead of $1$ in $xy +t=1$ ? Atiyah book

I have some confusion in Atiyah Commutative algebra Here is an outline 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. Proof :Let $x\in…
jasmine
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Question about zero-divisors and a quotient of a polynomial ring by an ideal in the book Introduction to commutative algebra by Atiyah and Macdonald.

I am reading the book the book Introduction to commutative algebra by Atiyah and Macdonald. I have two questions On Page 51. On Line 5 of Page 51, it is said that the zero-divisors in $A/\mathfrak{q} \cong k[y]/(y^2)$ are all the multiples of $y$.…
LJR
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Question about primary decompositions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On page 50, Line -7, it is said that "if $f: A \to B$ and $\mathfrak{q}$ is a primary ideal in $B$, then $A/\mathfrak{q}^c$ is isomorphic to a subring of…
LJR
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Localization of a PID $R$ is always a DVR. What about the converse?

Suppose $R$ to be a PID and let $M$ be a maximal ideal of $R$. It is well-known that the localization $R_M$ is a valuation ring. Furthermore, it is Noetherian. Then $R_M$ is a discrete valuation ring. Now, my question is the following: if we assume…
TheWanderer
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Doubt in the proof of Krull's principal ideal theorem

Krull's principal ideal theorem states that $A$ is a Noetherian ring and (x) is a principal, proper ideal of $A$, then each minimal prime ideal over (x) has height at most one. According to proof which is available on Wikipedia, it starts in the …
User
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