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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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Finite presentation of R-modules in an exact sequence

I have trouble proving the following question. Suppose we have an exact sequence $L\to M \to N \to 0$ of $R$-modules, with $M $ finitely presented and $L$ finitely generated. Show that $N$ is finitely presented. Previous result: Suppose we have…
Joost van Geffen
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If $R$ is a domain but not a field and $Q=Frac(R)$, prove that $Hom_{R}(Q,R)=\{0\}$.

I thought this problem is really easy and give the following 5 line proof: We assume $R$ is commutative as the book assumed. Suppose we have a non-zero homomorphism $\phi$, then $\ker(\phi)$ is a proper ideal of $Q$. But $Q$ is a field and has only…
Bombyx mori
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Vanishing degree on exact sequence

Let $0\rightarrow N \rightarrow M\rightarrow P \rightarrow 0$ be an exact sequence of finitely generated $\mathbb{N}$ graded module over a commutative ring $R$. The vanishing degree of a $\mathbb{N}$ graded module $M$ is defined to be the maximal…
Axy
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Is there a 'maximal prime ideal' contained in a given ideal?

It is well known, that every ring (commutative and unital) which is not the zero ring has at least one maximal (consequently prime) ideal. Let $R$ be a commutative and unital ring and $I\neq 0$ an ideal. Does there exist at least one maximal prime…
user8463524
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Derivations of a local algebra over a field

Let $A$ be a local algebra over a field $k$. Let $\mathfrak{m}$ be the unique maximal ideal of $A$. Suppose the canonical homomorphism $k \rightarrow A/\mathfrak{m}$ is an isomorphism. Let $f \in A$. There exists a unique $c \in k$ such that $f…
Makoto Kato
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Is $K[x_1, x_2,...]$ normal or not?

As the title speaks for itself, is the polynomial ring in infinitely many variables over a field normal or not? Can someone provide a reference/proof? Thanks Also, what about a directed union of normal subrings? Is that normal? Tangential to this,…
Dquik
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Do Euclidean rings admit $b$-adic representations ?

Let $(R,f)$ be an Euclidean domain with Euclidean function $f: R\setminus \lbrace 0 \rbrace \to \mathbb{Z}_+$. Given a fixed non-unit $b$ and $x \neq 0$, we obtain by division with remainder $x_0,y_1$ s.t. $x=x_0 + y_1b$. Repeating this process we…
Ralph
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Local Rings as Triples

A local ring is a (say, associative, commutative and unitary) ring $R$ with a unique maximal ideal $\mathfrak{m}$, which in turn determine a uniquely a field $k = R/\mathfrak{m}$. And then my book (and i've seen this in other places as well) says…
Thiago
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direct limit of rings?

In Atiyah's commutative algebra, he says in the intro that the word ring will always be commutative and has unity in the book. But in the exercise 21 in the chapter 2, a direct limit of a family of rings is introduced, which has no unity if the…
Mathcho
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How to show that the projective dimension is infinite

How can we show that the projective dimension of the $\mathbb{Z}/p^2 \mathbb{Z}$-module $\mathbb{Z}/p \mathbb{Z}$ is infinite?
user7475
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Krull's theorem and uniqueness

Commutative rings with unit must have a maximal ideal by Krull's theorem. But is it true, in general, that such sets must have a unique maximal ideal? Does it matter if the ring is finite or infinite?
Squirtle
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Pullback of Ring Spectra

let $\phi: R \to A$ a ring homomorphism and $f:Spec(A) \to Spec(R)$ the induced morphism on Specs. Let $p \in Spec(R)$. So $ R_p$ is local. This induces following commutative diagramm: $$ \require{AMScd} \begin{CD} Spec(A_p) @>{a} >> Spec(A)…
user267839
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Quotient of k-algebra finite-dimensional

Suppose $k$ is a field and $A$ is $k$-algebra and $\mathfrak{m}$ a maximal ideal. Consider the $k$-vector space $A/ \mathfrak{m}^N$ for some $N \geq 2$. Can we show that this is finite-dimensional? For $A = k[X_1,\ldots,X_n]$ this is true. But in…
JSchoone
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Normalization of $k[t^2,t^3]$

I am trying to prove that, for some field $k$, $k[t]$ is the normalization of the ring $k[t^2,t^3] \cong k[x,y]/(x^2 - y^3)$. I trying to do it as follows: Consider $R = k[t^2,t^3]$ and its normalization $\bar{R}$. Then, I want to show that…
urpi
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Direct proof of classification of $\textrm{Spec}(k[X])$, $k$ an algebraically closed field

Let $k$ be an algebraically closed field. Combining the Nullstellensatz, the fact that $\textrm{Dim } k[X,Y] = 2$, and Krull's height theorem, one can show that the prime ideals of $k[X,Y]$ consist of $(0), (X-a,Y-b)$, or $(f)$ for $f \in k[X,Y]$…
D_S
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