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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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on the proof of a simple inequality in dimension theory

Let $(R,m)$ be a local ring and $M \neq 0$ a finite $R$-module. Let $x \in m$ and set $\bar{M}=M/xM$. Then $\dim M/xM \ge \dim M -1$. One way to see this is as follows: let $\dim M/xM = s$ and let $y_1,\dots,y_s$ be a system of parameters for…
Manos
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Nilradical strictly smaller than Jacobson radical.

In a preparation question for an exam, I am asked to give an example of a ring $A$ such that the nilradical $\operatorname{Nil}(A)$ is strictly smaller that the Jacobson radical $J(A)$. Here's how I solved the problem: It is enough to find some ring…
Daniel Robert-Nicoud
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Spectrum of a localized ring

I must find the Spec of the localized ring k[x,y] at the ideal (x,y). I know that the spec of localized ring k[x] at (x) is {(0),(x)}. Is there any similar attribute?
georgeson1960
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Prove that these are primary decompositions.

Prove that $$\langle 4,2x,x^{2} \rangle=\langle 4,x \rangle\cap \langle 2,x^{2} \rangle $$ $$\langle 9,3x+3 \rangle=\langle 3 \rangle\cap \langle 9,x+1\rangle$$ are two primary decomposition in $\mathbb{Z}\left [ x \right ]$. I put my answer…
kpax
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Roots of Units in Complete $\mathbb{C}$-Algebras

$\newcommand{\cc}{\mathbb C}$ Let $R$ be a finitely generated $\cc$-algebra and ${\frak m}\subset R$ a maximal ideal. Denote by $\hat R$ the completion of $R$ with respect to $\frak m$. Assume that $x\in R^\times$ is a unit and let $\hat x$ denote…
Jesko Hüttenhain
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Write $\mathbb{R}[x]/(x^5+x^3)$ as direct product of its localizations

Let's consider the commutative ring $\mathbb{R}[x]/(x^5+x^3)$. We have that $x^5+x^3=x^3(x^2+1)$. So $\mathbb{R}[x]/(x^3(x^2+1)) \simeq \mathbb{C}[x]/(x^3) $. How can I write the artinian ring $\mathbb{R}[x]/(x^5+x^3)$ as direct product of its…
ArthurStuart
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$\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$

Let $\mathbb F_p[[T]]$ be the ring of formal series over $\mathbb F_p$, and $l\in\mathbb N.$ How to prove that: $\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$ ?
Med
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proving $(M\otimes N)/J (M \otimes N) \cong M \otimes \left(N/JN \right)$

Let $f: R \rightarrow S$ be a ring homomorphism of Noetherian rings, $M$ finite $R$-module and $N$ finite $S$-module which is flat over $R$. Let $J$ be an arbitrary ideal of $S$. I want to prove that there exists a canonical isomorphism $(M\otimes…
Manos
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An easy question (I think) about the subrings of $S^{-1}R$

Let $S$ be a multiplicative subset of a commutative ring $R$. Now consider the homomorphism $\phi_S :S^{-1}R \mapsto R$ where $\frac{r}{s} \mapsto r$ for any $s\in S$. Now my question is: Does this homomorphism create an inclusion preserving…
user53970
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graded homomorphism of finitely-generated, free modules with minimality condition

The present question is a follow up of this question: Finite free graded modules and the grading of their duals. Let $k$ be a field, $S=k[x_1,\dots,x_n]$ and $\phi: F \rightarrow G$ a graded homomorphism of graded, finitely-generated, free…
Manos
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minimal injective resolution and finiteness

Let $M$ be a finite module over a Noetherian ring $A$. Let $0 \rightarrow M \rightarrow I^1 \rightarrow I^2 \rightarrow \cdots $ be a minimal injective resolution of $M$. Question: Is it true that each $I^k$ is a finite $A$-module? If yes, how can…
Manos
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Characteristic Polynomial Computation

Let $A_{0}=k[x,y]$, $\mathfrak{m}=(x,y)$. Let $A=(A_{0})_{\mathfrak{m}}$. We wish to compute the characteristic polynomial, $\chi_{\mathfrak{q}}$, of the $\mathfrak{m}$-primary ideals (i) $(x,y)$, (ii) $(x,y^2)$, (iii) $(x,y)^2$ and finally check…
user 3462
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Hilbert Basis Theorem applied to integral domains

Was reading a solution to an exercise of the Atiyah-MacDonald "Introduction to commutative Algebra" and this passage catched my attention "Let $A$ be an integral domain, so by H-B-T we can infer that $A[x]$ is an integral domain too" I looked at…
Riccardo
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Basic Issue With the Hom Functor on Commutative Rings

In the category of $A$-modules, one has the following property: if $f:M \rightarrow M''$ is a map of $A$-modules, and the induced map $f^*:Hom(M'',N) \rightarrow Hom(M,N)$ is injective for all $N$, then $f$ itself must be surjective. The proof I…
Cass
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$(A+B)(A\cup B)=A+B$, where $A,B$ are ideals?

EDIT: Let $A$ be an ideal $S$ be any subset of commutative ring $R$. Define $AS$ to be of the form $\sum{as}$ for every $a\in A, s\in S$. Then $AS$ is an ideal. This is easy to see. Let $A,B$ be ideals of commutative ring $R$. I'm trying to…
user67803
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