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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.

16857 questions
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Some generating set of modules

In a module, we know what a minimal generating set is. But, is it always true that such a set exists? If the module is finitely generated, is it possible?
Sang Cheol Lee
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Alternative proof or typo?

In Atiyah-MacDonald, we have the claim that $S^{-1}(I+J) = S^{-1}I + S^{-1}J$ and similarly, $S^{-1}(IJ) = S^{-1}I S^{-1}J$. Here $I,J$ are ideals of a commutative unital ring $R$ and $S$ is a multiplicative subset of $R$. The proof is omitted with…
Rudy the Reindeer
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Examples of rings of fractions

I wanted to come up with a few examples of rings of fractions $S^{-1}R$. Can you tell me if these are correct: 1.Let $R = \mathbb Z$, $S = (2 \mathbb Z \setminus \{0\}) \cup \{1\}$. Then every $[x] = \frac{r}{s} \in S^{-1}R$ consists of the…
Rudy the Reindeer
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How to represent an ideal lattice using a matrix?

In Craig Gentry's thesis on implementing a homomorphic encryption scheme, he defines an ideal lattice as an ideal in the quotient ring $\mathbb Z[x]/\langle f\rangle$, with $f$ a polynomial of degree $n$. I understand that such a quotient ring can…
Andrew D'Angelo
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Is the integral closure of a local domain in a finite extension of its field of fractions semi-local?

Is the integral closure of a local domain in a finite extension of its field of fractions semi-local? If the answer is negative, I wonder under what conditions it would be semi-local. EDIT Here's an example of a local domain which is not…
Makoto Kato
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A proposition on a Dedekind domain

I need a proof of the following proposition(?). Actually I think I came up with a proof. But it's nice to confirm it and/or to know other proofs. Thanks. Proposition Let $A$ be a Dedekind domain. Let $I$ and $J$ be non-zero ideals of $A$. Then there…
Makoto Kato
  • 42,602
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On the proof of Theorem 4.3.2 in Bruns&Herzog (Part II)

Let $S=k[x_1,\dots,x_m]$ be the polynomial ring over a field $k$ and let $I$ be a homogeneous saturated ideal of $S$. Let $h$ be a linear form of $S$ such that it is $S/I$-regular. Consider the exact sequence \begin{align} 0\rightarrow I(-1)…
Manos
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Finding the tensor product of two finitely generated $R$-modules, where $R$ is a PID

I was asked to write down what the tensor product of two finitely generated $R$-modules $M,N$ is over a commutative ring $R$, which is a PID. I know that if $f \in R$, then $M \otimes_R R /\langle f \rangle \cong M / fM$, by considering exact…
Paul Slevin
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Is the completion of a local ring local?

Let $A$ be a noetherian local ring with maximal ideal $m$ and let $I\subset m$. Is the $I$-adic completion of $A$ necessarily local?
user93417
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Meet of all the powers of any prime ideal is 0.

Let $R$ be a commutative integral domain (with 1). I am looking for a general condition under which for all prime ideals $P$, the meet $\bigcap P^n$ of all the powers of $P$ is $0$. This is true for noetherian domains [Zariski \& Samuel, IV, Cor.…
Michael Barr
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Prime ideal generated by $\operatorname{height}P$ elements

Let $P$ be a prime ideal of height $n$ in a local ring $R$, which is generated by $n$ elements, say $P=(a_1,...,a_n)$. The image $\bar P$ of $P$ in $R/(a_1,...,a_i)$ , $1≤i≤n$, is an $(n-i)$-generated prime. I want to be sure that it is of height…
karparvar
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Why do there is a unique continuous homomorphism?

Is this a right place to ask help for an exercise? Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $n$ is invertible in $A$,…
user1678880
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Integral domain and ascending chain condition proof

Show that an integral domain $A$ is a principal ideal domain if every ideal $I$ of $A$ is principal, that is, of the form $I=(a)$. Show directly that the ideals in a PID satisfy the a.c.c.
inasse amasha
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Dimension of local ring by a zero-divisor

Let $(A,m)$ be a Noetherian local ring and let $x \in m$. If $x$ is a non-zero divisor, then we know that the Krull dimension satisfies $\dim A/(x) = \dim A-1$, see e.g. Atiyah-MacDonald Corollary 11.18. But what if $x$ is a zero-divisor? By…
Manos
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Reference for the determinant of an endomorphism of a projective module of finite rank

In Bourbaki algèbre commutative first book exercice 9 of paragraph 5 of chapter II (page 174) there is an exercise where they explain how to define the determinant of an endomorphism of a projective module of finite rank. I'm wondering if there is a…
anton
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