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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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Relationship between definitions of saturation of ideals

I am getting a little confused about the relationship (if any) between two definitions/uses of the term “saturation.” Let $R$ be a commutative ring with 1 and let $I\subset R$ be an ideal. Atiyah-MacDonald gives one characterization. If $S\subset R$…
Kenanski Bowspleefi
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Why is every element of a finite atomic lattice the meet of the set of meet irreducible elements over it?

I am having trouble understanding one argument in the proof of theorem 58.6 of "Graded Syzygies" by Irena Peeva. The theorem is Let $L$ be a finite atomic lattice. There exists a monomial Ideal whose lcm-lattice is $L$ At the end of the proof, we…
Roni Varshavsky
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Is this local homomorphism injective?

Let $f:(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism of local rings (i.e. the image of the maximal ideal $\mathfrak{m}$ is contained in $\mathfrak{n}$). Suppose that $B$ is a local regular ring, i.e. $\dim B =\dim_{B/\mathfrak{n}}…
mathfan24
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Systems of parameters for a $K$-algebra

I don't know how to solve the next problem: If we have two systems of parameters, $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ for a finitely generated $K$-algebra $A$ which is also an integral domain, and know that $A$ is free…
user87369
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Spectrum of a ring: contraction of prime ideals, preimage and quotient ring

Let $\psi:R\rightarrow S$ be a ring homomorphism. Let $\text{Spec}(\psi):\text{Spec}(S)\rightarrow\text{Spec}(R),\mathfrak{p}\mapsto\psi^{-1}(\mathfrak{p})$, where $\mathfrak{p}$ is a prime ideal of $S$. In fact, $\psi^{-1}(\mathfrak{p})$ contains…
user823
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Algorithm for finding generators of an ideal

Let $k$ be a field, and $f:k[x_1,\ldots,x_n]\to k[y_1,\ldots,y_m]$ a $k$-algebra homomorphism. Given $r_1,\ldots,r_k\in k[y_1,\ldots,y_m]$, is there an algorithm for producing a finite generating set for the ideal $f^{-1}((r_1,\ldots,r_k))$?
Julian Rosen
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I-adic topology on a ring

Given a commutative ring $R$ with proper ideal $I$, we can let $N_r=\{r+I^n\}_{n\in\Bbb N}$ and $\mathcal{B}=\bigcup_{r\in R} N_r$, and take the smallest topology containing these sets. I believe this is the $I$-adic topology? Am I correctly…
AlgebraQuestion
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Localisation and completion of $\mathbb{Z}$

This question might seem a little vague, but any information will be helpful. Fix a prime $p$ in $\mathbb{Z}$. It is easy to see as rings that the localisation at $(p)$ is contained in the $(p)$-adic completion (i.e.…
Coherent Sheaf
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What is meant by a "general linear form"?

I've been reading some papers lately on the topic of generic initial ideals and related stuff, and here and there the concept of a "general linear form" (or general quadric, quintic, etc.) comes up. This seems to have some special meaning in the…
beep27
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When does the generating set of a finite rank $A$-module reduce to a basis of the $A/\mathfrak{m}$-vector space?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and let $k = A/\mathfrak{m}$ be the residue field. Let $M$ be a finitely generated $A$-module; so $M/\mathfrak{m}M$ is a finite dimensional $k$-vector space. From Atiyah and MacDonald…
Hamish
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Zero-divisors and R-sequences

Here is an exercise which I can not solve. You may find it in Commutative Rings by Irving Kaplansky, p. 103, ex. 13. Let $R$ be a commutative ring with $1$ (not necessarily Noetherian) and $A$ an $R$-module. Let $x_1,...,x_m$ be an $R$-sequence…
Ehsan M. Kermani
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Why A[[X]] is completely integrally closed if A is completely integrally closed?

I was reading exercise 9.5 of Matsumura's Commutative Algebra. The problem goes like this: Let $A$ be an integral domain and $K$ its field of fractions. We say that $x\in K$ is almost integral over $A$ if there exists $0\ne a\in A$ such that…
zjl
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Infinitely many primary decompositions of an ideal

Given a noetherian ring $R$ and an ideal $I$, we know that if the associated primes of $I$ coincide with its minimal primes (i.e. $\text{Min}(I)=Ass(I)$ ) , then there is a unique irredundant primary decomposition of $I$ and if $Min(I) \subsetneq…
Radu Titiu
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Localization of quotient of polynomial ring over integral domain

I'm now reading Eisenbud's, Commutative Algebra, p.132~133, Nullstellensatz, General form. My question is, Q. Let $R$ be a integral domain and $Q$ a prime ideal of $R[x]$, and $S:=R[x]/Q$. Let $0\neq b:=g+Q \in S$ be a nonzero element and $K$ the…
Plantation
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Property of an edge ideal

User fbakhshi deleted the following question: Let $G$ be a simple graph with finite vertex set $X = \{x_1,\dots, x_n\}$. The edge ideal of $G$, denoted by $I = I(G)$, is the ideal of $R=K[x_1,\dots,x_n]$ generated by all square-free monomials…
user26857
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