Start on Bland Problem 1, Problem Set 4.1: Generating & Cogenerating Modules

In summary: RMMMMM = A + BA, B \leq Mm \in Ma \in Ab \in Bm = a + bM = \Sigma N_\alphaN_\alpha \leq MMN_\alpha \le
  • #36
Very good, Peter, I knew you can do it !A very small comment, the last lines are a little bit confusing and not quite correct.
I would write that something like this

Let .

Then there are for all , such that .

Let , then and = y.

Therefore is an epimorpism.
 
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  • #37
steenis said:
Very good, Peter, I knew you can do it !A very small comment, the last lines are a little bit confusing and not quite correct.
I would write that something like this

Let .

Then there are for all , such that .

Let , then and = y.

Therefore is an epimorpism.

Thanks for all the help, Steenis ... thanks to you I understand a lot more than when I started working on the problem ...

Regarding the last few lines of the above proof I thought I'd write out explicitly what is going for the case

We have where ... and ... ...

... and so ... ...
Now to show or illustrate explicitly in the case of \Delta = \{ 1,2, \ ... \ ... \ , \ n \} how f is an epimorphism ... ( ... not meant to be a proof ... just an illustration ...)Assume such that ...

and ... such that

... ... ... ... ...

... ... ... ... ...

and ... such that Then ... such that ... ... since is a module ...

and then Now

That is where ...

So is an epimorphism ...Is that basically correct ...

Peter
 
  • #38
It is basically correct, but change:

Peter said:
Then ... such that ... ... since is a module ...

and then

Now

into:
Define then , because N is a module.

And
 
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