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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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An $R/\mathfrak p$-sequence is also an $R$-sequence

Let $R$ be a commutative Noetherian ring with $\mathfrak p$ a nilpotent prime ideal, that is $\mathfrak p^r=0$ for some positive integer. If $\oplus_{i=0}^{r-1}\mathfrak p^i/\mathfrak p^{i+1}$ is a flat $R/\mathfrak p$-module, how to prove that an…
nick
  • 631
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Question on tensor product with field

Let $A$ be a finitely generated $K$-algebra which has no zero divisors. Here $K$ is a field of characteristic $0$. Let $K\subset L$ an algebraic field extension. Now let $f: L\to E$ and $g: \textrm{Quot}(A)\to E$ be two homomorphisms to another…
Hans
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Nilradical that is a prime ideal

Let $R$ be a non-reduced commutative ring(not necessarily Noetherian) with unit. Let the nilradical $\mathcal{N}$ of $R$ be a prime ideal with the property that $\mathcal{N}^2=0$. Do we know about the structure of such rings or any kind of…
Zac
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Question about the proof of the going-up theorem of Cohen-Seidenberg.

Let $S$ be a subring of $R$ such that $R$ is integral over $S$. Let $P$ be a prime ideal of $S$ and $M=S-P$. Let $S_M$ be the quotient ring of $S$ and $R_M$ the quotient ring of $R$. Let $i: S \to S_M, j: R\to R_M$ be the natural maps and…
LJR
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Calculation of length of modules

I am trying to calculate the length of two modules: $k[x,y]_{(x,y)}/(y-x^2,y)$ and $k[x,y]_{(x,y)}/(y-x^2,x)$. The claim is that the former has length 2 and the latter has length 1. But I am not sure why this is true: the chain of ideals of…
MathChopper
  • 357
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Every prime ideal has finite height in a Noetherian ring

In Corollary 11.12 of Atiyah-Macdonald, it says that in a Noetherian ring every prime ideal has finite height. It seems to come directly from Proposition 11.10, which says if $A$ is a Noetherian local ring, $\dim A \leq d(A)$, where $d(A)$ is the…
Emily Williams
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Comparing two expressions of the completion of a ring

Let $A$ be a commutative Noetherian ring and $I=(a_1,\cdots,a_n)$ an ideal. Then the $I$-adic completion of $A$ is isomorphic to $A [[ x_1,\cdots,x_n ]]/(x_1-a_1,\cdots,x_n-a_n)$. Now let $e$ be an idempotent of $A$ and apply the above result to the…
Manos
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$I$-adic topology on a topological $k$-algebra $R$.

Let $k$ be a field. Let $k[[x]]$ be the completion of $k[x]$ with respect to the ideal $(x)\subset k[x]$. Let $R$ be a topological $k$-algebra that is complete for the $I$-adic topology induced by some ideal $I\subset R$. We define…
JanBakfiets1
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Converse of a result in a Noetherian ring involving height

Let $A$ be a Noetherian ring and $\mathfrak{p} \in Spec(A).$ If $x \in \mathfrak{p}$ then it is well-known that $ht_{A/(x)}(\mathfrak{p}/(x)) \leq ht_A(\mathfrak{p}) \leq ht_{A/(x)}(\mathfrak{p}/(x))+1$ where $ht$ stands for height. Moreover, if $x$…
Ehsan M. Kermani
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A characterization of a nilpotent endomorphism

I am reading Proposition 8, page 9, of the book LOCAL ALGEBRA by Serre. Consider that $A$ is a commutative ring with identity and noetherian and $M$ a finitely generated $A$-module. For each $x \in {A}$ we define the endomorphism $x_M: M…
Tomais
  • 509
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exactness of total ring of fractions

If I have an injection $A \to B$ of noetherian reduced rings. Does this in general induce an injection $$ Q(A) \hookrightarrow Q(B) $$ of total rings of fractions? In the proof of Lemma 2.6 (Greuel et. al.) they say this is clear but I don't see…
pyrogen
  • 384
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Minimal prime ideals of the support of a module

in a book that I am reading it is claimed (without proof) that all minimal prime ideals of the support of a finite module $M$ over a local Noetherian Ring forms a subset of the set of all associated primes, Ass $M$. Is this true in general of we…
Herr Warum
  • 149
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If every projective $A$ module is free, then the only idempotents are $0$ and $1$

I have to prove that if every projective $A$ module is free, then the only idempotents of $A$ are $0$ and $1$. I know that $A$ contains an idempotent implies $A \cong G \times H$, where neither $G$ nor $H$ are zero, but i can’t do anything after…
Harun rashid
  • 157
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When is the torsion submodule of a $k[x,y]$-module a summand?

Let $A=k[x,y]$ and $M$ be a finitely generated graded $A$-module. I want to know if the torsion submodule $T$ of $M$ is a direct summand. Apparently, Kaplansky, Irving: A characterization of Prufer rings shows that if every finitely generated…
Bubaya
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Is it a Projective R-module?

R is a commutative ring with unity. Let $\textbf{P}$ be a finitely generated projective $\textbf{R[t]}$-module. Then is $\textbf{P/$t^n$P}$ a finitely generated projective $\textbf{R}$-module? Now in my attempt I have desperately tried using the…
Divya
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