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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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Exercise using Nakayama Lemma

Let $A$ be a commutative unitary ring, $I$ an ideal contained in the Jacobson radical of $A$; let $M$ be an $A$-module and $N$ a finitely generated $A$-module, and let $u: M \to N$ be a homomorphism. If the induced homomorphism $\bar u: M/ IM\to…
Anirban Saha
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Definition of the Filtration of a Module

Let $R$ be a ring, and $M$ an $R$-module. What is the definition of a filtration of $M$? Is it simply a descending sequence of submodules $M_{1} \supset M_{2} \supset M_{3} \supset \dots \supset \dots$. Does it have to be countably infinite or can…
user100101212
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ring automorphisms preserving module structure

(All rings here are assumed to be commutative and unital) I have a rather naive question (it is naive since in general, I expect the problem to be very hard). Is it possible to determine when automorphisms of a ring preserve a module structure with…
Grisha Taroyan
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Direct sum of asssociated primes of a module

Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module. Denote the set of all associated primes of $M$ by $Ass(M)$. If $R= \oplus_{i=1}^{n} M_{i}$ where each $M_{i}$ is an $R$-module, then: $Ass(R) = \cup_{i=1}^{n} M_{i}$. Call…
user6495
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Can matrix factorizations be nonsquare?

$\textbf{Background}$ On page 49 of a 1980 paper by Eisenbud, he defines matrix factorizations. Let $R$ be a (unital) commutative ring. Given $x\in R$, Eisenbud defines a matrix factorization for $x$ as a pair of $R$-linear maps $(\varphi:F\to…
Anonymous
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Are isomorphisms as modules always isomorphisms as rings?

For example, I'm trying to prove that for any ring $R$ and any multiplicative subsets $U,V$ of $R$, we have $$U^{1}R\otimes V^{-1}R \simeq U^{-1}(V^{-1}R)$$ as rings. I found that they're isomorphic as $U^{-1}R$-modules. Is an isomorphism between…
K.A.
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A module is finite when its quotient is finite?

Let $A$ be a ring, $I$ an ideal of $A$ and $M$ an $A$-module. Suppose $A$ is complete and separated in the $I$-adic topology, $M$ is separated in the $I$-adic topology and $M/IM$ is finite over $A/I$.Then it's said that $M$ is finite over $A$. Here…
Yuyi Zhang
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Exercise about prime ideals in a polynomial ring

Are considered prime ideals $q_{1}\subsetneqq q_{2}\subsetneqq q_{3} \subseteq A[X]$. Could you show that $q_{1}\cap A\neq q_{3}\cap A$ ?
rle3791
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Non-monic minimal polynomial implies non-integral element?

In a ring extension $R\subseteq S$, in order to check whether an element of $S$ is integral over $R$, I wonder why it is enough to check the minimal polynomial? In a concrete example, I found that the minimal polynomial was not monic (which we…
mathismath
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Lemma 10.23 from Atiyah and Macdonald

I'm trying to understand lemma 10.23 from Atiyah and Macdonald. Lemma 10.23. Let $\phi:A\rightarrow B$ be a homomorphism of filtered groups, i.e. $\phi(A_{n})\subset B_{n}$, and let $G(\phi):G(A)\rightarrow G(B)$,…
JanBakfiets1
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A flat algebra over a DVR is connected if the reduction modulo maximal ideal is

Let $R$ be a DVR with a uniformizer $\pi$ and let $M$ be a finitely generated flat $R$-algebra. Assuming that $M\otimes R/\pi R$ is a connected ring, is $M$ connected as well? My geometric intution tells me this should be true but I do not know how…
user700841
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Local Noetherian ring is Artinian when power of maximal ideal is zero.

Let $R$ be a local Noetherian ring and $\mathfrak{m}$ its maximal ideal. I have shown that either $\mathfrak{m}^n\neq\mathfrak{m}^{n+1}$ or $\mathfrak{m}^n = 0$ and supposedly in this latter case $R$ is Artinian. But I am having a hard time proving…
Jarne Renders
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How to show $\bigcap_{n=0}^\infty I^n=0$?

Let $A$ be a commutative noetherian ring with unity, $I$ is an ideal contained in the radical of $A$, how to show $\bigcap\limits_{n=0}^\infty I^n=0$? I want to show $I \bigcap\limits_{n=0}^\infty I^n= \bigcap\limits_{n=0}^\infty I^n$, but I don't…
Born to be proud
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Question about degree of a Hilbert polynomial.

Let $A$ be a notherian semi-local ring and $m=rad(A)$, the Jacobi radical of $A$. An ideal $I$ is called an ideal of definition of $A$ if $m^n\subseteq I\subseteq m$ for some $n$. Now, for a finitely generated $A$-module $M$, we define the Hilbert…
Yuyi Zhang
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Question about locally nilpotent element on module

Let $A$ be a noetherian ring and $M$ an $A$-module. An element $a\in A$ is said to be locally nilpotent on $M$ if for any $x\in M$ there exists an integer $n>0$, s.t. $a^nx=0$. Put $$p=\{a\in A\mid a \text{ is locally nilpotent on } M\}.$$ Then $p$…
Yuyi Zhang
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