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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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injective hull of $k[x_1,...,x_n]/m$ where $m=(x_1,...,x_n)$

Is there any source in which I can find the exact relation between the injective hulls of $k[x_1,...,x_n]/m$ and $k[|x_1,...,x_n|]/m$ where $m$ is the maximal ideal $m=(x_1,...,x_n)$?
greenspider
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Classify left cyclic modules of a countable Noetherian ring

Here is the question: Let $R$ be a countable Noetherian ring, and let $C$ be the equivalence classes of left cyclic $R$-modules. Show that $C$ is countable. I started with countable fields (or division algebras): any cyclic $R$-modules will be…
nekodesu
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Spectrum of $R \otimes_{\mathbb Q} R$, where $R$ is a zero dimensional ring

Here is a classical theorem from algebraic geometry which might motivate what I'm about to ask. Let $k$ be an algebraically closed field. If $A$ and $B$ are finitely generated $k$-algebras, then $$(\mathfrak m, \mathfrak n) \mapsto \mathfrak m…
D_S
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The set of associated primes are finite

Let $R$ be a Noetherian ring and $M$ be a $R$ module. Suppose the set of associated primes of a module $M$ is finite, then does it necessarily mean that the support of the module $M$ is closed? Support of $M$ is defined to the set of the all primes…
dongrugose
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Nakayama's lemma's for projective modules over local rings.

Let $(R, m)$ be a local ring and $M$ a finitely generated projective $R$-module. Let $n$ be the number of elements in a minimal generating set of $M$. Then by Nakayama's lemma $M/m M$ any minimal generating set of $M/ m M$ as an $R$-module must…
Yuugi
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Gorenstein Rings

My question refers to a notation in following excerpt: https://stacks.math.columbia.edu/tag/0DW6 What is $A[0]$?
user267839
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$(M\otimes_A N)_B \cong M_B\otimes_B N_B$?

Let $A \rightarrow B$ be a homomorphism of commutative rings. Let $M, N$ be $A$-modules. We denote $M\otimes_A B$ by $M_B$. We regard $M_B$ as a $B$-module. Then $(M\otimes_A N)_B \cong M_B\otimes_B N_B$?
Makoto Kato
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Regular Local Ring of Formal Power Series Regular

Let $k$ a field and $R= k[[X_1,X_2, ..., X_n]]$ the regular local ring of formal power series with maximal ideal $(X_1,X_2, ..., X_n)R$. How to prove that $R$ is a regular ring? My attempts: $k$ is clearly regular, so firstly I tried to do it…
user267839
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Compact infinite Hausdorff space

What properties hold on a compact infinite Hausdorff space? I encountered this example in Atiyah-Macdonald on chain conditions: Let $X$ be a compact infinite Hausdorff space, $C(X)$ the ring of real-valued continuous functions on $X$. Take a…
nekodesu
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flat base change and finitely generated

Given that $R\to S$ is a flat local ring homomorphism of two Noetherian local rings. Then is $S$ always a finitely generated $R$-module? This question stems from a small detail in a proof I am currently reading, which asserts that given the above…
T C
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Double Quotient by Ideals in Local Rings

Suppose I have a local morphism of local noetherian rings $\phi:A\to B$, $m_A, m_B$ the maximal ideals. Then the $m_B$-adic completion of $B/m_AB$ is: $$(B/m_AB)\hat{}= \varprojlim(B/m_AB/m_B^n) $$ I wanted to know if there is a better way to write…
Serser
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The global dimension of the localization ring of a complete local ring at a prime ideal of its subring

Let $R\subseteq S$ be a finite ring extension such that $R$ and $S$ are both complete noetherian local domains. Let $\mathfrak{p}$ be a prime ideal of $R$ and denote by $\mathfrak{q}_1,\ldots, \mathfrak{q}_n$ all the prime ideals of $S$ that lie…
G.-S. Zhou
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Comparing localizations at prime ideals for a finite ring homomorphism

Let $A\subseteq B$ be two commutative noetherian local domains such that $A$ is complete (and regular), and $B$ is finitely generated as an $A$-module. Can we deduce from this condition that $$B_{\mathfrak{q}\cap A} = B_\mathfrak{q}$$ for any…
G.-S. Zhou
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Is $\mathrm{Hom}(E(R/P),E(R/Q))=0$ for distinct maximal ideals $P,Q$ if $R$ is commutative but not Noetherian?

Let $E_P=E(R/P)$ and $E_Q=E(R/Q),$ $E(X)$ being the injective envelope of $X.$ If $P,Q$ are prime and $R$ is commutative Noetherian, then $\mathrm{Hom}(E_P,E_Q) \neq 0$ if and only if $P \subseteq Q$ (e.g., Proposition 4.21 of "Injective Modules" by…
Chris Leary
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Faithful modules and integral extensions

Suppose $R\subseteq S$ is an integral extension, $S$ is a finite $R$-module. Is it true that $S$ is faithful $R$ module? (To show faithfulness we need to show $\operatorname{ann}_R(S)=0$).
dongrugose
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