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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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Power series substitution endomorphisms: corollary $7.17$ in Eisenbud's Commutative Algebra

It the corollary $7.17$ in the commutative algebra book of David Eisenbud, as follows: I could not understand the highlighted line. What does mean by the line $\text{image of $\varphi(x)=f$ of the generator $X$ of ${\color{blue}{(x)}}$ of…
MAS
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Why $\cap \frak{n}^j=0$?

From David Eisenbud's commutative algebra book (page $199)$. I have some understanding problem in the proof of Theorem $7.16$ as follows: My questions: $(1)$ What type elements in $S$ ? Are they necessarily power series over $R$? See the…
MAS
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Exercise in Atiyah (localization)

Let me refer you to: http://www-users.math.umd.edu/~karpuk/chap3solns.pdf Page 2, ex. 4 Can you please explain the following step: $tb=f(s')b=s'b$ Why $f(s')=s'$ ? Thanks
user6495
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Show that $\mathfrak{a}^n\not\subset\mathfrak{q}$.

Let $\mathfrak{a},\ \mathfrak{p}\subset R$ be two ideals, $\mathfrak{p}$ prime, with $\mathfrak{a}\not\subset\mathfrak{p}.$ Show that $\mathfrak{a}^n\not\subset\mathfrak{p}$. I tried this by induction on $n$. So $n=1$ is just the hypothesis, supose…
Somerandommathematician
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Integral closure of ideal definition

I'm seeing what appear to be differing definitions of the integral closure of an ideal, and I want to know if they are actually different and what the accepted definition is. We have $A \subset B$ commutative rings and I an ideal of A. The first…
Flowers
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Why $y(1-x), z(1-x), x$ is not a regular sequence in the ring $\mathbb{C}[x,y,z]$?

Why order is an important matter in the following case of regular sequence? The following lines are mentioned in https://en.wikipedia.org/wiki/Regular_sequence $x, y(1-x), z(1-x)$ is a regular sequence in the polynomial ring $ \mathbb{C}[x, y, z]$,…
MAS
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Question about definition of colength.

Given a commutative ring $R$ over an algebraically closed field $k$, an ideal $I$ is of colength $n$ if $\operatorname{dim}_k(R/I)=n$. Then it’s said when $R=k[[x,y]]$, every ideal of colength $n$ in $R$ contains $(x,y)^n$. I am confused here. Just…
Yuyi Zhang
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If for ideal I, radical R(I) = I, is I a prime ideal

We know that for any prime P, the radical R(P)=P. However is the converse of this Statement true. That is, if we know that radical of an ideal I is itself, i.e. R(I)=I, is I prime? I presume it is not but couldn't come with a counterexample.
Anirban Saha
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finding out the mapping of the subring inside another ring

I was reading the book on Commutative Algebra by Miles Reid and during the discussion of the zero divisors the authors make the following comment. The ring $K[X,Y]/(XY)$ is a subring of $k[X] \oplus k[Y]$ with X and Y mapping to a non zero…
user2714795
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Localization of Localization isomorphic to localization, prime ideals

Let $R$ be a commutative ring, and $P,Q \subset R$ prime ideals such that $ P \subseteq Q$. Let $R_{P}$ denote the localization of $R$ at $P$. Prove that $R_{1} = R_{P}$ is isomorphic to $R_{Q}$ localized at $PR_{Q}$, denote this ring as $R_{2}$. I…
user100101212
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Proof of Weak Nullstellensatz by using maximal ideals?

I'm working through Atiyah and Macdonald, and I'm stuck on Chapter 1 Problem 27. It asks us to prove that for algebraically closed $k$, we have that all maximal ideals of the ring of polynomials in $n$ variables over $k$ are expressible as…
Michael Gintz
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Integral Noetherian ring over Artinian ring is Artinian

I'm struggling with one of the exercises in my course. It goes something like this: let $A$ be an Artinian ring (that is any descending chain of ideals stabilizes) and let $B$ be a Noetherian ring which is integral over $A,$ then I have to show…
Grisha Taroyan
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equivalent properties in absolutely flat rings

let A be a ring , I shall prove that : ($ Nil(A)=0 $ and Every prime ideal of $A$ is maximal) $\Rightarrow$ ($A_m$ is a field for each maximal ideal $m$ ) where $ Nil(A) $ is the nilradical of A now reading a proof somewhere , it said : All…
Laassila souhayl
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The height of the ideal generated by linear polynomials

Let $I=(f_1,\dots,f_r)\subset k[x_1,\dots,x_n]$ be an ideal with generators homogeneous linear polynomials and furthermore we assume that $f_1,\dots,f_r$ form a minimal generating set of $I$. Is the information above enough to tell that the height…
No One
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Proof of a equation about fibre of flat morphism

Let ($A,m,k$) and ($B,n,k'$) be noetherian local rings, and let $A\to B$ be a local homomorphsim. Let $M$ be a finite $A$-module and $N$ be a finite $B$-module which is $A$-flat. Then depth$_B(M\otimes_A N)$=depth$_A M$ + depth$_{B}(N\otimes k)$ In…
Yuyi Zhang
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