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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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Does $I^{-1}$ invertible imply $I$ invertible?

Now that I have your attention, here are the pertinent definitions: Let $R$ be an integral domain with field of fractions $K$. A $R$-fractional ideal of $K$ is a $R$-submodule $I$ of $K$ such that $aI\subseteq R$ for some $a\in R\setminus0$. From…
Matemáticos Chibchas
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Tensor product and injective maps

during a class we met a map $\Phi : \mathbb{Z}^n\to \mathbb{Z}^n$. We saw that $\Phi\otimes 1_\mathbb{Q}$ was an isomorphism and then the Professor said that it implied that $\Phi$ was injective. Unfortunately, I am not sure about the reason for…
John N.
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Example of a non invertible fractional ideal

I'm giving a seminar talk on the ideal class group, and am looking for an example of a fractional ideal that is not invertible. Does know of a simple example in say $k[x,y]$ or $\mathbb{Z}[\sqrt{-3}]$. Such examples must exist but computation there…
TheNumber23
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Does Localization Commute with Direct/Inverse Limits

Let $A$ be a ring and let $M_n$ be $A$-modules. For a prime ideal $P$ in $A$ is it true that $$(\varprojlim_n M_n)_P=\varprojlim_n (M_n)_P\text{ and } (\varinjlim_n M_n)_P=\varinjlim_n (M_n)_P?$$ If it is not true, are there conditions on $A$ and…
Sang Cheol Lee
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The integral closure of a finite separable field extension of the fraction field is finitely generated

Let $A$ be Noetherian and integrally closed in its field of fractions $K$ and $L$ a finite separable field extension of $K$. Why is the integral closure $B$ of $A$ in $L$ finitely generated over $A$?
Peter Patzt
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Is the $\mathfrak m$-adic completion of a radical ideal again a radical ideal?

Let $(R,\mathfrak m)$ be a local (Noetherian) ring and let $\hat{R}$ be its $\mathfrak m$-adic completion. Let $I$ be an ideal of $R$ which is a radical ideal. Is it then also true that $\hat{I}$ (the $\mathfrak m$-adic closure of $I$ in…
Sebastian
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Maximal ideals in rings of polynomials

Let $k$ be a field and $D = k[X_1, . . . , X_n]$ the polynomial ring in $n$ variables over $k$. Show that: a) Every maximal ideal of $D$ is generated by $n$ elements. b) If $R$ is ring and $\mathfrak m\subset D=R[X_1,\dots,X_n]$ is maximal ideal…
user82494
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Does every Krull ring have a height 1 prime ideal?

Let $A$ be a Krull ring. According to Theorem 12.3 in Matsumura's Commutative Ring Theory, the family of localizations of $A$ at height 1 prime ideals of $A$ forms a defining family of $A$. Question: Why such family exists? In other words, why…
Manos
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What is an example of a radical of sum of ideals not being equal to the sum of radicals?

What is an example of the radical of a sum of ideals not equal to sum of the radical of the ideals?
ast
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Homomorphism extensions over finite ring extensions

Let $A \subset B$ be a finite ring extension. Fix a ring homomorphism $\nu : A \to \Omega$, where $\Omega$ is an algebraically closed field. I want to show there exists a non-zero homomorphism $v : B \to \Omega$ of $A$-modules. I want show that…
LinAlgMan
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Being maximal ideal follows from being a kernel

The ideal of all polynomials in $k[x_1,\ldots,x_n]$ with zero constant term is maximal (since it is the kernel of the homomorphism $k[x_1,\ldots,x_n]\to k$ which maps $f\mapsto f(0)$). I understand why it is a maximal ideal. But I don't get why…
jhadfjkh
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Localizations of Dedekind Domains are Discrete Valuation Rings

I am trying to prove the following implication, and can't seem to find my way around all the equivalent definitions of Dedekind domains and DVRs: I have a ring $R$ with the following properties: 1) $R$ is Noetherian. 2) $R$ is integrally closed. 3)…
Roy Ben-Abraham
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Inverse limit by example

I'm trying to understand inverse limits. For this I am looking at the example (mentioned in Atiyah-Macdonald, page 102): We start with the topological abelian group $G = \mathbb Z$ (endowed with the topology induced by $|\cdot|_p$) and observes that…
Rudy the Reindeer
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Recalling result of tensor product of polynomial rings

Let $k$ be a field (alg closed if you want). Now let $I_{i}$ be an ideal of $k[x_{i}]$ for every $i \in \{1,2,\ldots,n\}$. Is it always true that: $$k[x_1,x_2,\ldots,x_n]/ \langle I_1,I_2,\ldots,I_n \rangle \cong k[x_1]/I_1 \otimes_k k[x_2]/I_2…
user10
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How to show a ring is normal or not, and how to show the normalisation of the ring

I am confused about how to show whether a ring is normal or not. For example, consider the $k$-algebra $k[x,y] /\langle x^2 - y^3 \rangle$, which is a domain. How do I show it is not normal? Are there any standard techniques? I know that I want to…
Paul Slevin
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