1
vote
0answers
49 views

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?
0
votes
0answers
42 views

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 ...
1
vote
1answer
46 views

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. ...
1
vote
1answer
51 views

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.
1
vote
1answer
18 views

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?
2
votes
1answer
236 views

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 $a\in A$. $f(a)\in Im\,\,f$, ...
2
votes
1answer
51 views

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 ...
3
votes
1answer
180 views

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 ...
2
votes
1answer
82 views

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 ...
2
votes
2answers
101 views

What is a non-degenerate module?

I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$) I came across a module being called ...
3
votes
1answer
66 views

Motivation for a Particular Module Construction

In Advanced Modern Algebra, Rotman gives the following construction of a module that he denotes by $V^T$: Let $V$ be a vector space over a field $k$ and let $T:V \rightarrow V$ be a linear operator ...
11
votes
1answer
451 views

Why do we call it trace?

For any module $P$, we define $\mathrm{tr}(P)=\sum \mathrm{im}(f)$, where $f$ ranges over all elements of $\mathrm{Hom}(P,R)$, and call it trace. Why does it have such a name? Does it have any ...
1
vote
0answers
65 views

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 ...
3
votes
2answers
182 views

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 ...