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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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Why does a minimal prime ideal consist of zerodivisors?

Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of…
Akhil Mathew
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Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1. In a ring $A$, let $\Sigma$ be the set of all ideals in which every element is a zero-divisor. Show that set $\Sigma$ has…
yunone
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Does localisation commute with Hom for finitely-generated modules?

Question. Let $R$ be a ring, $\mathfrak{p}$ a prime, $M$ a finitely-generated $R$-module, and $N$ any $R$-module. Is the natural map $$\textrm{Hom}_R(M, N)_\mathfrak{p} \to \textrm{Hom}_{R_\mathfrak{p}}(M_\mathfrak{p}, N_\mathfrak{p})$$ an…
Zhen Lin
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A non-noetherian ring with all localizations noetherian

If for a ring $A$ every localization $A_\mathfrak{p}$ by a prime $\mathfrak{p}\subseteq A$ is noetherian, is it true that $A$ is noetherian? I believe not but I can't find a good counterexample.
Daniele A
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Inverse limit of modules and tensor product

Let $(M_n)_n$ be an inverse system of finitely generated modules over a commutative ring $A$ and $I\subset A$ an ideal. When is the canonical homomorphism $$\left(\varprojlim\nolimits_n M_n\right)\otimes_A A/I \rightarrow \varprojlim\nolimits_n…
Cyril
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Do localization and completion commute?

Let $A$ be a commutative ring and $\mathfrak{p}$ be a prime ideal of $A$. Under which assumptions for $A$ and $\mathfrak{p}$ does localization by $\mathfrak{p}$ and completion with respect to $\mathfrak{p}$ commute? To be more precise, when is…
user7475
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Atiyah-Macdonald Exercises 5.16-5.19

I have solutions to Exercises 5.16–5.19 in Atiyah–Macdonald's Introduction to Commutative Algebra, but not in the order desired; I find myself needing later exercises to do earlier ones, and it's been frustrating me. Online solution sets (I count…
jdc
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Definition of a finitely generated $k$ - algebra

In Miles Reid's Undergraduate Commutative Algebra he defines a ring $B$ to be finite as an $A$ - algebra if it is finite as an $A$ - module. Now what I don't understand is suppose we look at the polynomial ring $k[x_1,\ldots,x_n]$ where $k$ is a…
user38268
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Does quotient commute with localization?

Let $R$ be a commutative ring, and $I \subset R$ an ideal. If we choose an element $x \in R$ we can consider $(R/I)_x$ and $R_x/I_x$. In general, does localization commute with quotient? i.e. $(R/I)_x \simeq R_x/I_x$? If not... are there hypotheses…
ArthurStuart
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When is the completion of a ring a local ring ?

Let $R$ be a commutative ring with unit and let $m$ be a maximal ideal of $R$. Are there known conditions on $R$ or $m$ such that the $m$-adic completion $\hat{R}$ of $R$ is a local ring. Since the completion of a Noetherian local ring is again…
Ralph
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Tensor product of a module with an ideal is isomorphic to their standard product

Let $A$ be a commutative ring and $M$ an $A$-module. Let $I$ be any ideal of $A$. We have an epimorphism $M \otimes_A I \rightarrow IM$. It seems to me that this is not in general an isomorphism. Q1: Any counterexample? If $M$ is flat, then $M…
Manos
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Is the radical of an irreducible ideal irreducible?

Fix a commutative ring $R$. Recall that an ideal $I$ of $R$ is irreducible if $I = J_1 \cap J_2$ for ideals $J_1$ and $J_2$ only when either $I = J_1$ or $I = J_2$. Question : Assume that $I$ is an irreducible ideal. Must the radical of $I$ be an…
Mary
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Atiyah-Macdonald, Exercise 8.3: A finitely generated $k$-algebra is Artinian iff it is a finite $k$-algebra.

Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent: (1) $A$ is Artinian. (2) $A$ is a finite $k$-algebra. I have a question in the proof of (1$\Rightarrow$2): By…
Gobi
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In a principal ideal ring, is every nonzero prime ideal maximal?

Inspired by this question, I was wondering whether from just the hypothesis that $A[X]$ is a nontrivial (commutative) principal ideal ring (so without supposing it is a domain) one can deduce that $A$ is a field. One possibility to prove that would…
Marc van Leeuwen
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Jacobson radical equal to nilradical in $R[X]$

Let $R$ be a non-zero commutative ring with identity. Let $\textrm{nilrad}(R)$ be the nilradical of $R$, which can be characterised either as the intersection of all prime ideals of $R$, or as the ideal of nilpotent elements. Let $J(R)$ be the…
Katie Dobbs
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