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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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Why does every maximal ideal closed in $\mathfrak{a}$-topology imply that $\mathfrak{a} \subseteq \text{Jac}(A)$?

I must be missing something very simple, but suppose that every maximal ideal $\mathfrak{m}$ of a Noetherian ring is closed in the $\mathfrak{a}$-topology on $A$. Then why does this imply that $\mathfrak{a} \subseteq \text{Jac}(A)$, the Jacobson…
David Benjamin Lim
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An example of an $m$-primary ideal in noetherian local domain

Is there any example of a $m$-primary ideal $I$ in a noetherian local domain $(R, m)$ such that $I^2=mI\not=m^2 $?
alinaghi
  • 91
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Ideal consisting of zero divisors

Let $I$ be a finitely generated ideal of a commutative ring $R$. Assume every element of $I$ is a zero divisor. Does then exist a $x \neq 0$ in $R$ with $xI=0$? This is true if $0$ is a decomposable ideal, for example if $R$ is noetherian. I wonder…
Martin Brandenburg
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A wrong proof about Dedekind domains

I "proved" that a Dedekind domain is a PID, but as we know this is wrong (for example $\mathbb{Z}[\sqrt{-5}]$). I do not know what is wrong in my proof: Suppose $R$ is a Dedekind domain, $I$ is any nonzero ideal of $R$, $\mathfrak p_i$ all the…
Strongart
  • 4,767
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The fixed subalgebra of a finitely generated algebra

Let $k$ be a field, $A$ a finitely generated $k$-algebra, put $A^{G}:=\{a \in A \mid g(a)=a ~ \mbox{for all}~g \in G\}$, where $G$ is a finite group of automorphisms of $A$. If (1) the order of $G$ is not divisible by the characteristic of $k$,…
Sang Cheol Lee
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Extensions and contractions of prime ideals under integral extensions

Let $R\subseteq S$ be an integral extension of commutative rings with identity. Let $P$ be a prime ideal in $R$ and $Q$ a prime ideal in $S$. If $Q=PS$ and $P=Q\cap R$ what can we say about $Q^n\cap R$? This ideal always contains $P^n$, but when…
BMI
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Inclusion containing V and radicals

Let $A$ be a commutative ring with unit and $I,J$ be two ideals of $A$. Also, denote $V(I):=\{\mathfrak{p}\in\operatorname{Spec}A\mid I\subset\mathfrak{p}\}$. Why is it true that if $J\subseteq \sqrt{I}$ then $V(I)\subseteq V(J)$? I guess it has…
studying
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Does $1 \otimes_A b= b\otimes_A 1$ imply $b \in A$?

Supppse $B$ is a faithfully flat $A$-algebra, and $b$ an element in $B$. Does $1 \otimes_A b= b\otimes_A 1$ in $B\otimes_A B$ imply $b \in A$?
user93417
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$\mathbb Q+X\mathbb R[X]$ is not Noetherian

Let $A=\{q+r_1X+ \cdots +r_nX^n: q \in \mathbb{Q}, r_i \in \mathbb{R}\}$ be the polynomial ring with rational costant terms. I have to prove that $A$ isn't a noetherian ring. How can I prove it?
ArthurStuart
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Reference about $k$-algebra isomorphism of $k[x_1,\ldots,x_n]$

$k$ is a field, we can assume $k$ is algebraic closed if we need. I donot know if the construction of $\mathrm{Aut}_{k-alg}k[X_1,\ldots,X_n]$ is known. I donot know any references about this group $\mathrm{Aut}_{k-alg}k[X_1,\ldots,X_n]$. What do we…
wxu
  • 6,671
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Can an integral domain be embedded in a proper quotient of itself?

Does there exist an integral domain $R$ which has a proper ideal $J$ so that there exists an injective ring homomorphism $\phi \colon R \to R/J$? If yes, what are suitable assumptions on $R$ to exclude such a behaviour? Especially, if $R$ is a…
Sebastian
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Free resolution of $(x^2, y^2, xy+yz)$

Let $S = k[x,y,z]$ ($k$ field) and let $I$ be the ideal $(x^2,y^2,xy+yz)$. I computed a minimal free resolution of $S/I$, and the dimensions of the free modules in the resolution are 1,3,3,1. (Just to make sure, I confirmed the result with…
Ted
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Generating a regular sequence out of two

Here is the last problem of my final exam in "Commutative algebra" which I think, no one has solved it completely, today! Let $R$ be a commutative Noetherian ring. Let $a_1,\dots,a_n$ and $b_1,\dots,b_n$ be two regular sequences in $R.$ Prove that…
Ehsan M. Kermani
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Does $\operatorname{Spec}(B/\mathfrak{p}^e)=\operatorname{Spec}((A/\mathfrak{p}) \otimes_A B)$?

This is from Atiyah-Macdonald Ex7.23. (It seems that the entire problem is not needed, but I will write it just in case: Let $A$ be a Notherian ring, $f:A \to B$ a ring homomorphism of finite type. $f^*:\operatorname{Spec}(B) \to…
Gobi
  • 7,458
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Primary decomposition example

I want to find the primary decomposition of $(x^2, xy^2)$ as an ideal of $k[x,y,z]$ where $k$ is some field. My guess is $(x^2, xy^2) = (x) \cap (x^2, y^2)$ however I am not 100% certain if $(x^2, y^2)$ is a primary ideal. My approach to see this…
Craig
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