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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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Noetherian Local Ring

I came across this old exam problem. If $R$ is a local Noetherian ring and $I$ is an ideal in $R$ such that $I^2=I$ then $I =0$. Any hint would be appreciated. I'm only familiar with what the definition of local and Noetherian mean. I'm not sure why…
Mykie
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If $A\rightarrow B$ be faithfully flat ring homomorphism, then $IB\cap A=I$

Let $A,B$, be rings, $I\subset A$ an ideal, and $A\rightarrow B$ be faithfully flat ring homomorphism. Show that $IB\cap A=I$. By our assumptions, I know that for any $A$-module $M$, we have an injection $M\rightarrow M\otimes_A B$, given by…
Mictej
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Why is $\langle x^2, xy,y^3\rangle$ primary in $k[x,y,z]$?

Can someone tell me a quick reason as to why $\langle x^2, xy,y^3\rangle$ is primary in $k[x,y,z]$? I'm trying to read a solution and I don't get this.
Bombi
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What is the dimension of $R[[X]]$ where $R$ is a Noetherian ring?

If $R$ is a Noetherian ring then we know that the Krull dimension of the polynomial ring $R[X]$ is $\rm dim(R)+1.$ Is there any formula for the Krull dimension of the power series ring $R[[X]] $ When $R$ is a Noetherian ring ?
user371231
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Exercise 2.13 in Atiyah-Macdonald

Let $f \colon A \to B$ be a ring homomorphism, and let $N$ be a $B$-module. Regarding $N$ as an $A$-module by restriction of scalars, form the $B$-module $N_B = B \otimes_A N$. Show that the homomorphism $g \colon N \to N_B$ which maps $y$ to $1…
Earthliŋ
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How to show that a ring is not discrete valuation ring?

Suppose $K$ is a field and $R=K[x,y]/(y^2-x^3)$. The question requires to show the localization at $(x,y)$ is not a discrete valuation ring. I can find the unit element in the localization is the polynomial a with a non trivial constant term. The…
Honghao
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How to show a quasi-compact, Hausdorff space be totally disconnected?

This is from Atiyah-Macdonald. I was asked to show if every prime ideal of $A$ is maximal, then $A/R$ is absolutely flat, Spec($A$) is a $T_{1}$ space,further Spec($A$) is Hausdorff. The author then asked me to show Spec($A$) is totally…
Bombyx mori
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Modules $M$ such that the automorphism of $M \otimes M \otimes M$ induced by the permutation $(123)$ is the identity

I've been struggling with the following problem for a couple of days and I don't seem to get any further: Let $R$ be a commutative ring. I would like to get (something like) a classification of all finitely generated $R$-modules $M$ that satisfy the…
Sebastian
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Example of an application of a theorem about ideals in rings of fractions in Atiyah-MacDonald

In Atiyah-MacDonald, we have the following theorem (p. 41): Proposition 3.11. i) Every ideal in $S^{-1}R$ is an extended ideal. ii) If $I$ is an ideal in $R$ then $I^{ec} = \bigcup_{s \in S} (I : \langle s \rangle )$. Hence $I^e = (1) = S^{-1}R$ if…
Rudy the Reindeer
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An extension with the induced map on Spec being bijective.

Let $A$ be a commutative ring with unit. Let $A\subset A[b]$ be an extension of rings such that $b^n, b^m\in A$, where $m,n$ are positive integers that are coprime with each other. Show that $SpecA[b]\to SpecA$ is bijective.
messi
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General Form of $S^{-1}A$ - modules

I have been trying to show that if a ring $A$ is absolutely flat then so is the localisation $S^{-1}A$ by any multiplicative set. Now while trying to do this, I asked myself the following: Is there a description for the general form of an $S^{-1}A$…
user38268
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Atiyah-Macdonald, Exercise 5.4

I was having some trouble with the following exercise from Atiyah-Macdonald. Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Let $\mathfrak{n}$ be a maximal ideal of $B$ and let $\mathfrak{m}=\mathfrak{n} \cap A$ be the…
user135520
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A ring with ACC on prime ideals whose spectrum is non-noetherian.

I am currently working on the converse of the exercise #12 on chapter 6 of Atiyah-Macdonald's book on commutative algebra. The problem is asking whether there is a ring $A$ which satisfies the ascending chain condition on prime ideals (that is,…
Scream
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Localization at a prime and direct limits

Let $R$ be a commutative ring with $1 \neq 0$ and let $P \subset R$ be a prime ideal. Apparently we have $$\varinjlim\limits_{f \in R \setminus P} R_f \cong R_P$$ where $R_f$ the the localization of $R$ at the set $\{1,f,f^2,\ldots\}$. Why is this…
nigel
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Are there any integral domains in which no nonzero prime ideal is finitely generated?

Are there any integral domains in which no nonzero prime ideal is finitely generated? (Other than fields, of course, where the condition is vacuously satisfied.) I asked a similar question the other day, but the solution there relied on using…
Daniel McLaury
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