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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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Definition of P-primary ideal and symbolic power

The following is definition of them, from Eisenbud, Commutative algebra with a view toward algebraic geometry. $M$ be any module over ring $R$ and $P$ be a minimal prime ideal over the $\mathrm{ann}(M)$. Then the submodule $M'$ of $M$ defined by…
user673940
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Minimal generating set of a module exists?

I'm reading Matsumura's Commutative Ring Theory Thm7.10, which states Let $(A,m,k)$ be a local ring, $M$ be a flat module over $A$ and $x_1,\cdots,x_n\in M$. If the images $\overline{x_1},\cdots,\overline{x_n}\in M/mM$ are linear independent over…
nessy
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If $R_{f_i} \rightarrow S_{f_i}$ is of finite type ,where $(f_1, \dots , f_n)=(1)$ ,then $R \rightarrow S$ is of finite type

Statement : If $R_{f_i} \rightarrow S_{f_i}$ is of finite type for $i=1,2, \dots,n$ such that $(f_1, \dots , f_n)=(1)$ ,then $R \rightarrow S$ is of finite type. [QUESTION] Is that true? I think It is not true (see below motive) but I can't find…
hew
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One dimensional noetherian domain with every maximal ideal being principal: any radical ideal is principal

Let $R$ be a one-dimensional Noetherian domain such that every maximal ideal is principal. If $I$ is a radical ideal ($\sqrt I = I),$ show that $I$ is principal. Since $R $ is a domain, $\{0\}$ is prime; so given $P$ prime ideal, since…
user2345678
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Are all prime ideals of $\mathbb C[x,y]/(y^2-x^3+x)$ maximal?

In fact I'm trying to prove that $\mathbb C[x,y]/(y^2-x^3+x)$ is a Dedekind domain. Till now I believe I was able to show that it is a Noetherian integral domain (easy) which is an integrally closed domain. If I prove that all prime ideals of…
Alexey
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Sharp's Exercise 16.33

I am reading Sharp's book "Steps in Commutative Algebra". Let $R$ be a commutative Noetherian ring and let $M$ be a non-zero finitely generated $R$-module. Let $a_1,\ldots,a_n \in R$. Suppose that there exist positive integers $t_1,\ldots,t_n$…
Busra
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Sharp's Exercise 16.37

I am reading Sharp's book "Steps in Commutative Algebra". But I really had difficulty in the exercise below. So can you give me a hint about the exercise below? Exercise 16.37 Let $R$ denote the ring of polynomials $\mathbb{Z}_{2\mathbb{Z}}[X]$ …
Busra
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Separated morphisms are stable under base change

Suppose that the map $f$ in the following diagram is a separated morphism (i.e. $\Delta_{X/S}:X\rightarrow X\times_{S}X$ is a closed immersion). I want to prove that $p_{2}$ is also a separated morphism. $$\require{AMScd}$$ \begin{CD} X\times_{S}Y…
Jozef
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$\mathbb{R}[X]$ is an integral extension of $\mathbb{R}[X^2-1]$

I am trying to prove that every polynomial of $\mathbb{R}[X]$ satisfies a monic polynomial equation with coeffients in $\mathbb{R}[X^2-1]$ that is every polynomial $b(x)= x^m+b_{m-1}x^m-1+...+b_{0}$ satisfies that there exits an $ n\in N$ such…
Vegan Maths
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Is $\mathfrak{b}^{ce} = \mathfrak{b} $ where $c$ and $e$ are contraction and extension of an ideal.

Let $f: A \rightarrow B$ be a ring homomorphism. They symbols $c$ and $e$ are contraction and extension of an ideal. One of the result says that $\mathfrak{b}^{ce} \subset \mathfrak{b} $. I feel that the equality should hold since $\mathfrak{b}^{ce}…
MUH
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How to prove $\operatorname{Ass}\operatorname{Hom}_R(M,N)=\operatorname{Supp}M\cap \operatorname{Ass}N$

If $R$ is Noetherian and $M$ and $N$ are finitely generated $R$-modules, show that $$\operatorname{Ass}\operatorname{Hom}_R(M,N)=\operatorname{Supp}M\cap \operatorname{Ass}N$$ where $\operatorname{Supp}M$ is the set of all primes containing the…
Attendre
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$k[x^2,x^3]/p$ ($p$:nonzero prime) is integral over $k$?

Let $p$ be a nonzero prime ideal of $A=k[x^2,x^3]$. I want to show $p$ is maximal. My trial is that $A/p$ contains $k$ and since $k$ is a field, if I can show that $A/p$ is integral over $k$ then it should be a field, too. But is it true that $A/p$…
Gobi
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Commutative Algebra: Quasi-Frobenius Implies Frobenius

Let $A$ be a commutative Algebra. Then quasi-Frobenius implies Frobenius, i.e. if $A$ is injective as a left module, then $_AA \cong {}_ADA$. My lecture notes only say this implication is true because $A$ is basic. It is easy to see that $A$ is…
Chaser01
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Quotient of Polynomial Ring is Artinian

I start with the polynomial ring $R = \mathbb{C}[x,y]$ and the ideal $I=(x^2 + ax, y^2 + by, xy + bx, xy +ay)$ for some $a,b \in \mathbb{C}^*$, $a\neq b$. I would like to prove that $R/I$ is artinian. I know that that $R$ and thus $R/I$ are…
EinStone
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$M\otimes _k S\cong M$?

If $0\rightarrow K\rightarrow S$ is an injective ring homomorphism of commutative rings and if $M$ is an $S$-Module am I right that $M\otimes _K S\cong M$?
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