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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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Proof of Theorem 4.2.1 in Herzog-Hibi, "Monomial Ideals"

The Theorem and its proof can be found here. Specifically, i am stuck at the fourth paragraph of the proof. Let me give some context: Let $I$ be a graded ideal over a polynomial ring $S=K[x_1,\dots,x_n]$ over a field $K$ and consider a monomial…
Manos
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Necessary and sufficient condition for a regular sequence.

$f_1, \ldots, f_r$ is a regular sequence in $S/I$ (where $S$ is a polynomial ring in $n$ variables, and $I$ its ideal) iff $$(I, f_1, \ldots, f_{i-1}): (f_i)= (I, f_1, \ldots, f_{i-1}) \quad i \ge 2.$$ I think a sequence is regular if $f_1 + I…
user337578
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determinant annihilates ring vector

Let $R$ be a ring, and let $A$ be an $n\times n$ matrix with coefficients from $R$. Suppose for $r\in R^n$ we have $Ar=r$. Prove that $\det (A-I)\cdot r=0$. It is actually part of a bigger problem where $r_i$ generate ring $B$ which is module finite…
ralleee
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What is the Hilbert Series of $R/I$ for a regular sequence?

What is the Hilbert series of $R/I$ for $I = (F,G)$ where $F,G$ is a regular sequence on $R = k[x,y]$ with $\deg F \leq \deg G?$ Definition: A sequence $F,G$ is regular on $R$ if $F$ is a nonzero divisor of $R$ and $G$ is a nonzero divisor of…
jacob smith
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Question about proof of chain conditions

Here is a proof from Atiyah-Macdonald: For i) $\implies$ ii) could one not write "If $(x_n)$ is such that $x_m = x_{m+1} = \dots$ then obviously $x_m$ is a maximal element"? I am asking because the book has been getting terser with proofs and has…
Rudy the Reindeer
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Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself?

Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself? This is the converse of if the initial ideal of an ideal is radical, then the ideal is radical. Also, how can I create more…
Mark
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Another question about a proof in Atiyah-Macdonald

I have a question about the following proof in Atiyah-Macdonald: 1:Why is $\Omega$ infinite? Are all algebraically closed fields infinite? 2: How does the existence of $\xi$ follow from $\Omega$ being infinite? Thanks.
Rudy the Reindeer
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Nil but not Nilpotent

Is there a commutative ring $R$ with zero Krull dimension such that its Jacobson radical is nil but not nilpotent? Of course, in Noetherian case (which leads to Artinian case) for $R$ each nil ideal is nilpotent, so the ring, if it exists, should…
karparvar
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Noether Normalization, finiteness over a sub algebra

I'm currently doing an exercise on Noether Normalization in the context of a course on commutative algebra and I'm not sure whether the solution I have come up with is correct or does even make sense. Reason for that is probably also, that I'm…
Sebastian Hebold
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Injective homomorphism $R^n\rightarrow R^m$ implies $n\leq m$?

Let $R$ be a nonzero commutative ring, and let $n,m$ be integers. Let $f:R^n\rightarrow R^m$ be an injective homomorphism of $R$-modules. I'm trying to show that $n\leq m$. My idea: Assume that $n>m$, let $i:R^m\rightarrow R^n$ be the "embedding in…
Elinor Rautner
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Need help with the proof that set of monomials $N$ belonging to a monomial ideal $I$ is a $K$-basis for $I$.

The proof given in the text I'm reading is (here $S=K[x_1, \dots , x_n]$): The bit I'm having problems with understanding is how does $v=u_i w$ imply that supp$(f) \subset N$? It would make sense to me if the implication were that supp$(f)\subset…
Mark
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Germs of continuous functions

Let $R$ be the ring of germs of continuous functions $\mathbb R \rightarrow \mathbb R$. It is clear that this is a local ring with maximal ideal $\mathcal m$ consisting of those germs $f$ with $f(0)=0$. What is not clear to me is why $(\mathcal m)^2…
Cyril
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Example of an integral domain with a non-principal prime ideal of height one

Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of height $1$ which is not a principal ideal?
user8463524
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A condition which every module (over a commutative noetherian ring) having finite length should satisfy

Let $G$ be a module over the non-trivial commutative Noetherian ring $R$. Show that if $G$ has finite length then there exist an ideal $M$, which is a product of finitely many maximal ideals of $R$, such that $MG=0$
pritam
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Why is the natural map $I^{-1}J\rightarrow\operatorname{Hom}_R(I,J)$ epi?

I am reading Eisenbud’s book Commutative Algebra, and have gotten stuck on the proof of Theorem 11.6c. Let $R$ be a ring, $K(R)$ its total quotient ring, and $I,J\subset K(R)$ invertible $R$-modules (i.e. locally free of rank one). Then the claim is…
Tomo
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