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Nov
9
comment A ring with IBN which admits a free module with a generator with less elements than a basis
@t.b. There it is, in page 11! Great. As expected, the example is quite nasty :) Thank you once again, Theo. You can perfectly post is as an answer, I'll accept it if in a safe time from now no one offers another (perhaps simpler?) example.
Nov
9
comment Cardinality of a minimal generating set is the cardinality of a basis
Georges: just to let you know, here's a followup to this question, math.stackexchange.com/questions/80658/…
Nov
9
asked A ring with IBN which admits a free module with a generator with less elements than a basis
Nov
9
accepted Cardinality of a minimal generating set is the cardinality of a basis
Nov
9
comment Differential forms on fuzzy manifolds
This question should be more upvoted: what a great amount of detail and information! Reading this question is interesting in itself.
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
Georges' answer below made me realize that giving a generator with $s$ elements of a module $M$ is the same as giving an epimorphism $R^s\to M\to 0$. In light of this, my question is actually this one: math.stackexchange.com/questions/20178/… ! Because of the different formulation, I hadn't found it before asking the question; I wouldn't have asked the question if I had ;P
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
Great! Yes, I'm aware of the restriction-extension of scalars constructions, but I'm not yet familiar with their applications. What an useful and elegant application! Thank you once again.
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
Thank you for the clarification; I was misunderstanding the argument above, I thought it used that $k^s$ and $k^r$ were finite dimensional. Please, correct me if I'm wrong: what you have done is to "extend scalars" from $R$ to $k$, right? $k$ is a $k$-module and hence an $R$-module through restriction of scalars with the canonical map $R\to k$; and we have that $k\otimes_R R^n$ is a $k$-module (extension of scalars).
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
Thank you, it's nice to know. It still puzzles me that facts regarding basis of free modules behave a little better in the infinite case!
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
Thank you for your answer. Could you elaborate, though, on yout last line? Also, what if $r$ is infinite?
Nov
7
comment Cardinality of a minimal generating set is the cardinality of a basis
@JoelCohen Sorry, I missed that definition. I edited it in.
Nov
7
revised Cardinality of a minimal generating set is the cardinality of a basis
added 70 characters in body
Nov
7
asked Cardinality of a minimal generating set is the cardinality of a basis
Nov
5
asked Is the axiom of universes 'harmless'?
Oct
31
comment Expanding $(2y-2)^2$ by FOIL
What on earth is FOIL???
Oct
29
revised Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
I edited the title since it was not at all descriptive and hence made the question difficult to find
Oct
29
suggested approved edit on Global invertibility of a map $\mathbb{R}^n\to \mathbb{R}^n$ from everywhere local invertibility
Oct
27
comment Measure of reals in $[0,1]$ which don't have $4$ in decimal expansion
Just in case, the "infinite monkey principle" is usually called Borel-Cantelli lemma.
Oct
26
comment Must a solution defined for $(t_0,+\infty)$ with bounded limit in $+\infty$ tend to a constant solution?
Oh, indeed. I believe this is the correct formulation of my comment: "If a function with constant limit in infinity has a derivative with existing (& finite) limit in infinity, then this limit must be zero."
Oct
26
comment Every $R$-module is free $\implies$ $R$ is a division ring
@Pete: I second Amitesh's comment, for sure, and I thank you for a wonderful set of notes! I didn't know the noncommutative algebra notes; I looked up the argument there and expanded the answer (which, in fact, was a bit confusing, as I myself was confused by the matter of sidedness). Thank you.