# Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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### Integrally closed factor rings and projective modules

I have a weird vision that comes from reading a paper by Raphael and Desrochers.. Let $R$ be commutative unitary semiprime ring such that for any integral and essential element $a$ of $R$, $R[a]$ is ...
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### Primary decomposition for modules

I am quite curious about the definition and applications of the primary decomposition for modules. The definition of a primary submodule. (Let's assume we work over a commutative noetherian ring $R$ ...
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### Maximal ideals in the ring of continuous real-valued functions on R

For a compact space $K$, the maximal ideals in the ring $C(K)$ of continuous real-valued functions on $K$ are easily identified with the points of $K$ (a point defines the maximal ideal of functions ...
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### An “Elementary” Math Question Generalized (Ring Theory Perhaps)

The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics" "Prove that if integers a_1, ..., a_n are all distinct, then the ...
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### Resources on Invariant Theory

Hi, So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd ...
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### are deformations of torsion modules always torsion?

Let's say I have a field $\mathbb{K}$ and a flat family of $\mathbb{K}[t]$-modules $M$ over the formal disk $Spec \mathbb{K}[[h]]$. Now, assume that $M/hM$ is torsion as a $\mathbb{K}[t]$-module (...
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### Atiyah-MacDonald, exercise 2.11

Let $A$ be a commutative ring with $1$ not equal to $0$. (The ring A is not necessarily a domain, and is not necessarily Noetherian.) Assume we have an injective map of free $A$-modules $A^m \to A^n$...
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### is there a good computer package for working with bicomplexes?

I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
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### What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "...
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### What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?