# Tagged Questions

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### Soft question (Etymology - Flatness)

Why where flat modules named "flat"? Is it because they are necessarily torsion free so in a "not convoluted" or circular like $\mathbb{Z}/n\mathbb{Z}$ is as a $\mathbb{Z}$-module?
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### Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
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### Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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### Definition of $p$-torsion module

Let $p$ be a prime. What is the definition of a $p$-torsion module ? I have googled but was not convinced.
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### Terminology for a type of indecomposable module

Let $R$ be a ring. Is there a name for an $R$-module that is indecomposable, and each of its quotients is also indecomposable?
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### A module as an external direct product of the kernel and image of a function

If $f:A\rightarrow A$ is an R-module homomorphism such that $ff=f$, show that $$A=Ker\,\,f\oplus Im\,\, f$$ Here is a part of what I made as a proof. Let $a\in A$. $f(a)\in Im\,\,f$, ...
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### characterization of certain idealizers of submodules

Consider two commutative rings with identity, $R$ and $S$ where $R$ is a principal ideal domain, $S$ is free and finitely-generated as an $R$-module and suppose there is a $R$-module homomorphism ...
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### Origin of the terminology projective module

Projective modules were introduced in 1956 by Cartan and Eilenberg in their book Homological Algebra. Does anyone know why they chose the word "projective"? Does it have something to do with the ...
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### What do you call this generalization of a module?

A module $M$ is an Abelian group with the extra property that any element $m \in M$ can be multiplied by any element $r$ in a ring $R$. I want to relax this definition so that the $M$ need not be an ...
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### Terminological question: generalization of rank to matrices over modules

I'm interested in what to call the length of the sequence of ones in the Smith normal form of a matrix. Equivalently (or so it seems to me), this is the rank of the largest submodule of M, on which A ...
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### Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...