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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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What is a "dual of a ring"?

As known, we can define dual vector space $V^{\vee}$ as $$V^{\vee}=\mathrm{Hom}_{k}(V, k)$$ where $V$ is $k$-vector space. Also, for finite abelian group $G$ we can define its Pontryagin dual $G^{\vee}$ as $$G^{\vee}=\mathrm{How}(G,…
Seewoo Lee
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Noetherian ring and primary decomposition result

I'm struggling with the following problem and I would appreciate some help if possible Let $R$ be Noetherian and let $I,J$ be ideals. Define $(I:J^{\infty}) = \bigcup_{n}(I:J^{n})$. (a) If $Q$ is primary, prove that $(Q:J^{\infty}) = Q$ for any $J…
Dquik
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finitely generated module over $R[U^{-1}]$ is the localization of a finitely generated module over $R$

This is Exercise 2.10 of Eisenbud's $\textit{Commutative Algebra}$: Show that every finitely generated module over $R[U^{-1}]$ is the localization of a finitely generated module over $R$. I'm not sure how to start because I don't see what a natural…
Massimo
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Is $\mathbb{C}[x]_{(x)}=\mathbb{C}[x]$?

I don't know what's the difference between $\mathbb{C}[x]_{(x)}$ and $\mathbb{C}[x]$. Isn't the localization is just equal to the original ring? Then why the first presentation is used?
Gobi
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When does $\mathfrak{a}B\cap A = \mathfrak{a}$?

Let $A\subset B$ be rings, and let $\mathfrak{a}$ be an ideal of $A$. Under what circumstances does $\mathfrak{a}B\cap A = \mathfrak{a}$? More precisely, are there conditions on $A,B$ that guarantee this for all ideals $\mathfrak{a}\triangleleft…
Ben Blum-Smith
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Which elements of a ring have zero differential?

Let $A$ be an algebra over $B$ (all rings commutative with a unit) and let $\Omega_{A/B}$ be the module of differentials of $A$ over $B$ with $d:A\to \Omega_{A/B}$ the universal derivation. Is there a nice characterization of the set $Z=\{a\in A |…
KotelKanim
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Noetherian domains with finitely many primes

For any domain $A$ let $A^\times$ be its group of units. Let $A$ be a noetherian domain with only finitely many prime ideals, and field of fractions $K$. Is the group $K^\times/A^\times$ finitely generated?
Pierre-Yves Gaillard
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A question about Cohen-Kaplansky domains

Let $A$ be a domain. Recall that $A$ is Cohen-Kaplansky (or CK) if (CK) any nonzero nonunit of $A$ is a product of irreducible elements, and there are only finitely irreducible elements up to multiplication by units. Consider the following…
Pierre-Yves Gaillard
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$I\cdot J$ principal implies $I$ and $J$ principal?

Let $R$ be a Noetherian domain, and let $I$ and $J$ be two ideals of $R$ such that their product $I\cdot J$ is a non-zero principal ideal. Is it true that $I$ and $J$ are principal ideals ? This seems an easy question to settle, but I can't find an…
Lierre
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Description of the set of prime ideals of the $R/m^2$

Let $R$ be a commutative ring and $m\subseteq R$ be a maximal ideal. Can you describe the set of prime ideals of the $R/m^2$. Are they all maximal ?
t.k
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Spectrum of finite $k$-algebras

Let $k$ be a field and $A$ be a finite $k$-algebra. How does one quickly see that $Spec(A)$ is a finite set? Further, is it true that the cardinality of $Spec(A)$ is equal to $dim_k(A)$?
Cyril
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Is a regular sequence ordered?

A regular sequence is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring $$ \frac R {\langle r_1, r_2, \cdots, r_{i-1}…
James Davidoff
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System of generators of a homogenous ideal

Let $I$ be a homogenous ideal in the ring $k[x_{1},\dots,x_{n}]$. My question is: If $\lbrace f_{1},\dots,f_{r}\rbrace$ is a minimal system of generators of $I$, then are the integers $r$ and $\deg f_i$ determined uniquely by $I$? More precisely:…
Arsenaler
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Atiyah and Macdonald, exercise 11.7

I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian. The very beginning poses a problem, he states in the hint that: for a prime of height $m$ we can choose $m$ elements in that prime such that the…
baltazar
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Dimension inequality for homomorphisms between noetherian local rings

$A$ and $B$ are commutative noetherian local rings with maximal ideals $\mathfrak{m}$ and $\mathfrak{n}$ respectively. If $f\colon A \to B$ is a local ring homomorphism, how do I prove the inequality $$\dim(B) \leq \dim(A) +…
Joni
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