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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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About Krull domain

If $A$ is a Krull domain in its field of fractions $F$, and if $F'\subset F$ is a subfield, then $A\cap F'$ is a Krull domain. But is it necessarily a Krull domain of $F'$? That is, is $F'$ necessarily the field of fractions of $A\cap F'$?
ashpool
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Are valuation rings coherent?

Serre wrote in his letter to Grothendieck(Oct. 25,1959) that valuation rings are coherent. How do you prove it?
Makoto Kato
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a proof in the book by Irving Kaplansky

Theorem 179 in the book Commutative Rings by Kaplansky states that: Let $R$ be an integral domain. The following conditions are necessary and sufficient for $R$ to be a UFD. (1) $R$ satisfies the ascending condition on principal ideals (2) In the…
Anh_Rose 1210
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Principal ideals in a commutative ring R

Given $A$ and $B$ principal ideals with the sum $A+B$ also principal. How to show $A\cap B$ is principal? If $A+B$ happens to be the unit ideal then I see that $A\cap B=AB$ which is principal. I tried deriving analogous properties with $A+B$…
Karthik C
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Understanding proof of a corollary leading up to Nakayama's Lemma

I would appreciate help on what should be an easy concept in the proof of a corollary leading up to Nakayama's Lemma. This link to mathoverflow.com (in the green highlighted section) gives the development as presented in "Atiyah and Macdonald" as…
user12802
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A commutative ring which has a group acting locally finitely on itself

The following definitions and proposition are motivated by this question. All rings are assumed to be commutative and have identity elements. Definition 1 Let $B$ be a ring. Let $A$ be a subring of $B$. An element $b$ of $B$ is called radical over…
Makoto Kato
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Calculation of Gröbner basis.

I'm not all that familiar with commutative algebra, so I need help with the calculation of a Gröbner basis. Let $ k $ be a field, and consider $ R = k[x_{1},x_{2},x_{3},x_{4}] $. I need to find the reduced Gröbner basis for the intersection of the…
Haskell Curry
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Noether normalization over $\mathbb{Z}$

I would like to know what is a correct analogue of Noether normalization theorem for rings finitely generated over $\mathbb Z$. Obviously, Noether normalization can not hold "literately" in this case since, for example the ring $\mathbb Z_2[X]$ does…
agleaner
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$\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated

Let $A$ be a commutative ring with 1, set $V(\frak a)=$ the set of prime ideals of $A$ that contains $\mathfrak a$, and write $\operatorname{Supp}(M)$ for the support of the $A$-module $M$. We always have $\operatorname{Supp}(M)\subseteq…
user221483
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Is a field extension $L/K$ finite if and only if $L$ is a finitely-generated $K$-algebra?

I recently started learning commutative algebra from Atiyah-MacDonald. This means that for the next few months, I'll be posting some (mostly silly) questions to check my understanding. (Thank you all in advance for your patience.) My…
Jesse Madnick
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Homogeneous ideal and degree of generators

Let $I$ be a homogeneous ideal in a graded local commutative ring $R$, $S$ be its minimal homogeneous system of generators. So, we know that the cardinality of $S$ is unique as the dimension of the vector space $I/\mathfrak{m}I$, where…
knot
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Product of fractional ideals

Let $R$ be a Noetherian commutative ring. Let $I,J\subset K(R)$ be fractional ideals where $K(R)$ is the total quotient ring. Define $I^{-1}:=\{s\in K(R) : sI\subset R\}.$ Further suppose that $I$ is invertible I.e. $I^{-1}I=R$. Then $I^{-1}J=\{s\in…
Tom
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Proper 2-generator ideal is not the intersection of 2 proper coprime ideals

Let $R$ be a polynomial ring (in finitely many, say 8, indeterminates) over an algebraically closed field $k$. Suppose $I=(f,g) \neq R$ is a proper ideal of $R$ and $I = J \cap L$ for two proper ideals $J,L \neq I$. How do I show $J+L \neq R$? I'd…
Jack Schmidt
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Example of duality behaving badly for modules on a finite length local $k$-algebra.

Let $R$ be some local $k$-algebra, which is finite dimensional as a vector space. (Eisenbud lists 0-krull dimension as an extra hypothesis in this section, but it seems redundant - I think it follows from Noether normaliation...) ($k$ is a field.) I…
Elle Najt
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Noetherian semiprimary rings

Is any Noetherian semiprimary ring $R$ Artinian? By semiprimary I mean $R/J(R)$ semilocal and $J(R)$ nilpotent, where $J(R)$ is the Jacobson radical of $R$. I know that if $R$ is Artinian then $J(R)$ equals the set $N(R)$ of nilpotent elements of…
karparvar
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