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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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Ideals of $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$

I was wondering if the ring $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}(\sqrt[3]{2})$ is a PID. I believe that it is because I think $\mathbb{Q}(\sqrt[3]{2}) \otimes_{\mathbb{Q}}\mathbb{Q}[x]$ is a PID, which are just polynomials with…
bob
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How to check if a ring is Artinian?

By definition, an Artinian ring is a ring that satisfies the descending chain condition on ideals. In practice, how to check if a ring is Artinian? For example, let $R$ be the quotient of the commutative ring $\mathbb{C}[x_1,x_2,x_3]$ the ideal $I$…
LJR
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At most one subring of fractional field is invertible ideal

Let $R$ be a domain and $K$ be fractional field. Let $I$ an invertible $R$-ideal. Show that $I$ is proper, i.e. $I:I=R$. Deduce that an additive subgroup $I \subset K = Q(R)$ is an invertible ideal for at most one subring of $K$. By using…
Desunkid
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Conditions/Counterexample for equality/lower bound in Krull height theorem

I am looking for a counterexample to the following version of the Krull height theorem. Let $R$ be a commutative Noetherian ring with $1$. Let $I := (x_1, \ldots, x_n)$ be a finitely generated ideal with minimal generating set $\{x_1, \ldots,…
G. Chiusole
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Zero divisors in polynomial rings with several indeterminants.

This question contains a characterization of zero divisors in $A[x]$ for a commutative ring $A$ with identity. But obviously the trick used in the proof does not work anymore for $A[x_1,...,x_n]$ mainly because whatever ordering we use for…
William Sun
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Why completion of a Noetherian ring is complete and Hausdorff

Let $R$ be a commutative Noetherian ring, $I$ is an ideal of $R$. It is well known that the completion of $R$ is complete and Hausdorff. In most books of commutative algebra, the proof is as follows: 1.$\hat R \otimes I\cong \hat I$;2. $\hat…
Jian
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Question about existence of a minimal resolution in the proof of Theorem 19.6 in Matsumura, Commutative Ring Theory

Let A be a local ring and M an $A$-module which has a finite free resolution $$0\rightarrow F_{1}\rightarrow F_{0}\rightarrow M\rightarrow 0.$$ In this situation, does a minimal resolution $$0\rightarrow L_{1}\rightarrow L_{0}\rightarrow…
C. Fujinomiya
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A map of algebras inducing a bijection on maximal ideals is a bijection

Choose an algebraically closed coefficient field $k$. Denote by $A$ a finitely generated integral domain over $k$. Let $f$ be a homomorphism of $k$-algebras $A\rightarrow A$ such that the pullback map from set of maximal ideals of $A$ to itself is a…
user693243
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Does a DVR of equal characteristic with a perfect residue field contain it?

Let $R$ be a DVR of equal characteristic with a perfect residue field $k$. Does there exist a unital homomorphism $k\rightarrow R$ such that the composition with the projection $R\rightarrow k$ is the identity? If the residue field is imperfect of…
user691994
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Classify prime ideals in the polynomial algebra over the power series ring

Is there a simple classification of the prime ideals in $k[[x]][y]$ for $k$ an algebraically closed field? This is a two-dimensional ring so we can divide all prime ideals according to their heights, possible values being $0$, $1$, $2$. Since the…
user692020
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Krull-Akizuki theorem for Noetherian reduced rings

Wikipedia article about Krull-Akizuki theorem gives a very general version of theorem: let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. If B is a subring of a finite extension L of K containing A and is not a field,…
user468985
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Show that a map is continuous in the Zariski topology

Let $R,S$ be two commutative rings with unity and $\alpha :R\to S$ be a ring homomorphism, for $f\in R$ is a non nilpotent element let $R_f$ denote the localization of $R$ with respect to the multiplicative closed set $\{1,f,f^2,f^3,\dots\}$, and…
user60184
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Conntectedness of SpecR

Let $R$ be a commutative ring with unity, $e\in R$ is called and idempotent if $e^2=e$ and if $e\notin \{0,1\}$ then it is called a non-trivial idempotent.want to show that $\text{Spec}R$ is not connected if and only if there exists a non trivial…
i.a.m
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Is there always an $R$-algebra with nontrivial Picard group?

Let $R$ be a nonzero commutative ring. Must there exist an $R$-algebra $S$ (that is, a commutative ring $S$ containing a quotient of $R$ as a subring) such that $\mathrm{Pic}(S)$ is not the trivial group? My first thought is that it suffices to…
diracdeltafunk
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Tensor a finite algebra with the residue field

Let $f:A\rightarrow B$ be a morphism of commutative unital rings. Assume that there exists a positive integer $n$ and a surjective map of $A$-modules $A^n\rightarrow B$. Let $\mathfrak{p}\subset A$ be a prime ideal with the residue field $k$. Then…
jon
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