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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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Krulls intersection theorem without Noetherian

Krulls intersection theorem states Let $R$ be a noetherian integral domain and $I\subset R$ an proper ideal. Then $\cap_{n>0}I^n=0$. What are some simple counterexamples if we forget the fact that $R$ is noetherian or an integral domain?
njlieta
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Show that $0 \to \mathbb{Q} \to \mathbb{R} \to \mathbb{R}/\mathbb{Q} \to 0$ split?

How to show that this sequence split? I'm trying to construct a map $\phi: \mathbb{R}/\mathbb{Q} \to \mathbb{R}$ by $\overline{r} + \mathbb{Q} \mapsto r$. Let the quotient map be $\pi$ and $\pi \circ \phi(\overline{r}) = \pi(r) = \overline{r}$ is…
nekodesu
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Is the direct limit of submodules also a submodule?

$\{M_i\}_{i\in I}$ is a directed system with direct limit $M$. For each $i\in I$, $N_i\subseteq M_i$ is a submodule and $\{N_i\}_{i\in I}$ with the restriction maps is also a directed system with direct limit $N$. So there is a natural map from $N$…
user150248
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Atiyah-Macdonalds Exercise 5.1

I'm attempting Atiyah-Macdonalds Exercise 5.1: Let $f: A \to B$ be an integral homomorphism of rings. Show that $f^{\ast}: \text{Spec}(B) \to \text{Spec}(A)$ is a closed mapping, i.e. that it maps closed sets to closed…
nekodesu
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Prove (projective$\Rightarrow$flat) using Lambek's criterion

Lambek's criterion states that (left) $R$-module $M$ is flat if and only if $M^{*} = \mathrm{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$ is an injective (right) $R$-module. Using this, I want to prove that every projective $R$-module is flat. To…
Seewoo Lee
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Prove $(R[X]/PR[X])_Q$ is a regular local ring

In an attempt to prove that polynomial ring of a regular ring is a regular ring, I encounter this beautiful result "$R$ is a local Noetherian ring. $Q\in \operatorname{Spec}(R[X])$ and $P=Q\cap R$. Then $R_P\to R[X]_Q$ is flat local and…
T C
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A question about the injective hull of the residue field

Let $(R, {\frak m})$ be a local ring and set $k=R/{\frak m}$. Over a Gorenstein local ring of dimension $n \geq 1$ we have $E_R(k)={\frak m} \, E_R(k)$. Is the relation true when $\dim R \geq 1$ and $R$ is not necessarily Gorenstein?
Mohammad Bagheri
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Is every subalgebra of $F[x]$ one-dimensional?

Let $F$ be a field. Let $A$ be a subalgebra of the polynomial ring $F[x]$. Does $A$ necessarily have Krull dimension $\leq 1$?
user15464
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Geometrically reduced algebraic extension of a field

Let $k$ be a field. Let $A$ be a commutative algebra over $k$. We say $A$ is geometrically reduced over $k$ if $A\otimes_k k'$ is reduced for every extension $k'$ of $k$. Let $K$ be an algebraic extension of $k$. It is well-known that $K$ is…
Makoto Kato
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Atypical definition of "regular ring"?

In my adventures I've finally come by a copy of Examples of Commutative Rings by Hutchins, and almost immediately read something that surprised me. On page 10, a regular ring is defined as Noetherian ring whose every localization at a maximal ideal…
rschwieb
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Localization of product of fields

Let $K$ be a field and $R = K\times K$. Let $P = 0\times K$ be a prime ideal of $R$. If $\phi : K \rightarrow R$ is defined by $\phi(x) = (x, x)$ then we need to prove the induced map $\phi_{P} : K_{0} \rightarrow R_{P}$ is surjective, where $K_{0}…
Rajesh
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Dedekind domain if and only if smooth manifold

I would appreciate help with the following. Let $k$ be a field that's algebraically closed and let $f$ be a polynomial in $k[X,Y]$. Prove that $R=k[X,Y]/(f)$ is a Dedekind domain if and only if at one of $f(a,b)$ and the two partials at $(a,b) \in…
Dquik
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Question in proving Nakayama's Lemma

There was a prior question regarding intuiting Nakayama's Lemma: Intuitive explanation of Nakayama's Lemma I am currently studying Reid's "Undergrad. Commutative Algebra." His statement of the lemma is specifically in the context of a local ring…
user12802
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An exact sequence from Koszul complex

Let $x$ denote the sequence $(x_1,\ldots,x_m)$ and $x^{'}$ denote $(x_1,\ldots,x_m,y)$. Then we have the following exact sequence of Koszul homologies: (set I=$(x_1,\ldots,x_m)$) $0\to H_1(x)/yH_1(x)\to H_1(x^{'})\to R/I\to R/I\to 0.$ Suppose the…
Jelly bee
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Any connection between $\operatorname{Ass}_A(M)$ and $\operatorname{Ass}_{A/α}(A/α\otimes_AM)$?

$A$ is noetherian. When taking quotient of $A$, is there something like taking localization: $\operatorname{Ass}_{S^{-1}A}(S^{-1}M)=\operatorname{Spec}(S^{-1}A)\cap \operatorname{Ass}_A(M)$ I’m studying primary decomposition. In this circumstance…
XiaYu
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