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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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A question about the statement of the Prime Avoidance Lemma.

The Prime Avoidance Lemma states the following: Suppose $I_1,I_2,\dots,I_n,J$ are ideals of a ring $R$, such that $J\subset \cup U_i$. If $R$ contains an infinite field, or if at most $2$ of the ideals $I_1,\dots,I_n$ are not prime, then $J\subset…
user67803
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Ideal of an ideal being an ideal itself

Take a commutative ring with unity (a.k.a. ring) $R$ and let $I$ be an ideal in $R$. Are the following true?? a) If $J$ is an ideal in $I$ (observed as a ring) , $J$ is not ideal in $R$(if this is true, please give me example) b) If $J$ is a maximal…
nikola
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If $A\subset B$ is faithfully flat and $B$ is finitely presented, is $A$ finitely presented?

Fix a field $k$. Does there exist a faithfully flat extension $A\subset B$ of $k$-algebras where $B$ is finitely presented but $A$ is not? Edit: This question has been put on hold for missing context. I am not sure what context will help the…
user577334
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How to localize $J$ at $a_{i}$?

Assume $I$ is a finitely presented $R$ module such that for $\langle a_{i}\rangle=(1),a_{i}\in R$, we have $I[\frac{1}{a_{i}}]=R[\frac{1}{a_{i}}]$. Define $J=Hom_{R}(I,R)$, what is a good way to show $$\displaystyle…
Bombyx mori
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The two definitions of Cohen-Macaulay ring

If $R$ is Noetherian local, then $R$ is CM iff $depth(R)=dim(R)$. If $R$ is only Noetherian, then $R$ is CM iff $R_P$ is CM for all prime ideals. I want to ask, if $R$ is local, does the two definitions coincide? The same question is true for…
T C
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A question about Jacobson ring

Let $A$ be a Jacobson ring and let $M$ be a maximal ideal of $A[X_1,...,X_n]$. Show that $M\cap A$ is a maximal ideal of $A$. We assume $A$ is commutative. Can someone give some tips for me? I can't solve this problem. Thank you.
张旭辉
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Why is the following map injective

I just have a quick question consider the map: $$\psi : \mathbb{C}[x,y] / (y - x^2) \rightarrow \mathbb{C}[t]$$ $$x \mapsto t$$ $$y \mapsto t^2$$ I want to check that this map is injective. In particular, I showed that it has inverse map. I…
user329017
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Normalisation of Quotient of Polynomial Ring

My question refers to a former thread of mine: Normalization of a Scheme Example The setting is the following: We have the ring $A=\mathbb {C} [X,Y]/(XY)$ and consider an arbitrary prime ideal $p$ of $A$. I want to verify that for each such ideal…
user267839
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If $M$ is flat over $A$ and and $mM\neq M$ for any maxideal $m$, then for any $N$, if $N\otimes_R M=0$, then $N=0$

I'm having trouble understanding the proof, for the following. Let $R$ be ring, $M$ an $R$-module. If $M$ is flat over $R$ and for any $m$ maximal ideal we get $mM\neq M$, then for any $R$-module $N$, if $N\otimes_R M=0$, then $N=0$. In the…
njlieta
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Field of fractions generated over a base field

Let $A$ be an integral domain and let $K$ field of fractions. Let $k$ be a subfield of $K$ such that $K$ is simple finite extension of $k$. Is it true that there is an element $ a \in A $ such that $ K = k[a]$?
Hodge-Tate
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Proving that $\dim(R[t]) \leq 2 \dim(R) + 1$ using localization

I'm studying for an exam, this is part of an exam question. Let $S = R[t]$ where $R$ is an integral domain. Let $Q < P$ be prime ideals of $S$ such that $Q \cap R = P \cap R = P_0$. Let $Y = R-P_0$ Show that $P_0S$ is a prime ideal of $S$ and…
pizzaroll
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If $R$ is a ring and $I$ and $J$ are ideals, then if $\operatorname{rad}(I) + \operatorname{rad}(J) = R$, then $I + J = R$

How can I show that if $R$ is a ring and $I$ and $J$ are ideals, then if $\operatorname{rad}(I) + \operatorname{rad}(J) = R$, then $I + J = R$? This has had me stumped for some time, which is frustrating as I imagine it has a fairly elementary…
Daven
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A zero-dimensional ring is Noetherian?

Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian. Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq m$ for a maximal ideal $m$. Hence $dim A \ge 1$,…
Manos
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Integrally closed domain equivalence

I'm trying to solve the next problem: Let $R$ be an integral domain, and $\lbrace S_i\rbrace_i$ a family of multiplicative closed subsets of $R$ such that $R=\bigcap_i S_i^{-1}R$. Then $R$ is integrally closed if and only if each $S_i^{-1}R$ is.…
NitroMaster
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Conditions under which finite length modules are artinian.

We already know that finitely generated modules over artinian rings have finite length but can anyone tell me under which conditions a finitely generated module of finite length is artinian? Thank you!
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