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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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every field of characteristic 0 has a discrete valuation ring?

How can we prove that every field of characteristic 0 has at least one Discrete Valuation Ring? My effort: Let $K$ be an field of characteristic 0. Then $\mathbb{Z}$ is a subring of $K$. Let $p$ be a prime. By Theorem 10.2 in Matsumura, there exists…
Manos
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How to show $M_{\mathfrak q}$ is flat over $A$

Let $f:A\rightarrow B$ be a homomorphism of commutative rings, and $M$ a finite $B$-module. If $a\in A$ and $M_a$ is a free $A_a$-module, then for a prime ideal $\mathfrak q$ of $B$ with $f(a)\notin\mathfrak q$, how to prove that $M_{\mathfrak q}$…
nick
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Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura

Let $I$ and $J$ be ideals of a Noetherian ring $A$. If $JA_P\subseteq IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subseteq I$. I'm reading Matsumura's Commutative Ring Theory book on my own. This is exercise $6.4$ in that book, and I…
Vladimir
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A question related to Krull-Akizuki theorem

Let $(R,m)$ be a D.V.R with field of fraction $K$ and $L$ any finite algebraic field extension of $K$. Suppose $\bar{R}$ is the integral closure of $R$ in $L$. Then it is well known that $\bar{R}$ is a Dedekind domain and for any nonzero ideal $J$…
A.G
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Is a regular ring a domain

A regular local ring is a domain. Is a regular ring (a ring whose localization at every prime ideal is regular) also a domain? I am unable to find/construct a proof or a counterexample. Any help would be appreciated.
Dev Bappa
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Automorphism group of a formal power series ring

Let $A$ be a commutative ring. Let $A[[x]]$ be the ring of formal power series in one variable. Can we determine the structure of the automorphism group of $A[[x]]$ over $A$? This is a related question.
Makoto Kato
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Prime ideals in an arbitrary direct product of rings

By ring I mean commutative unital ring. The prime ideal structure of a finite direct product of rings is well known: For $\prod_{i=1}^n R_i$, it is of the form $\prod_{i=1}^n P_i$ where only one $P_j$ is a proper prime ideal of $R_j$ and for $i\neq…
Andrew Chiriac
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In a faithfully flat ring extension, is $\operatorname{ht}I=\operatorname{ht}IS$ right?

For Noetherian rings $R$ and $S$, let $R\rightarrow S$ be a faithfully flat ring extension and $I$ an ideal of $R$. Does $\operatorname{ht}I=\operatorname{ht}IS$ hold? Is it a conclusion in some books on commutative algebra?
gaotian81
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Integral extensions: one prime lying over implies equal localization

Here's a problem from Matsumura's book "Commutative ring theory" page $69$. Let $A$ be a ring and let $A \subset B$ be an integral extension, and $\mathfrak{p}$ a prime ideal of $A$. Suppose that $B$ has just one prime ideal $P$ lying over…
user6495
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A question on faithfully flat extension

This question arose while reading page 116 of Red Book by Mumford. Let $B$ be a faithfully flat extension of $A$. Can I claim that $b \otimes 1 = 1 \otimes b$ in $B\otimes_A B$ if and only if $b\in A$?
A.G
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Isomorphism in localization (tensor product)

Let $A$ be a commutative ring with $1$ and let $M,N$ be $A$-modules. Since there is a map $f: A \rightarrow S^{-1}A$, defined by $a \mapsto \frac{a}{1}$ then given any $S^{-1}A$-module we can view it as a $A$ module via restriction of scalars…
user6495
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Is there a name for this commutative algebra property?

Let $A\subset R$ be rings such that every maximal ideal of $R$ contracts to a maximal ideal of $A$. Of course this is not always true, but it is the case for plenty of interesting examples, so it seems like this property might have a name. Does…
pw1
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What's the projective limit of these polynomial rings ?

Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m< n)$. Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ?
Tomasz
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Classification of prime ideals of $\mathbb{Z}[X]/(f(X))$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $A = \mathbb{Z}[X]/(f(X))$. Let $\theta$ = $X$ (mod $f(X)$). My question: Is the following proposition correct? If…
Makoto Kato
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Possibly false proof in AM

Here is the excerpt of the book where I suspect a mistake (page 66): Where they say "The restriction to $A$ of the natural homomorphism $A^\prime \to k^\prime$" I think we don't want a restriction. We start with the quotient map $\pi: A[x^{-1}] \to…
Rudy the Reindeer
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