Torsion Modules & Finite Generation: Investigating the Connection

[pivoxa15]

Homework Statement

Does a torsion module M imply M is cyclic? Or does it imply M is finitely generated?

I think cyclic implies torsion module. What about the reverse?

The Attempt at a Solution

I think there is a connection but don't see it.

 

[matt grime]

Just construct counter examples to show that neither of those implications is true.

Cyclic does not imply torsion, torsion does not impliy cyclic, torsion does not imply finitely generated, finitely generated does not imply torsion, finitely generated does not imply cyclic.

The only thing that is true is that cyclic implies finitely generated.

 

[pivoxa15]

What would a torsion R-module M over a PID R imply?

We can't use the structural theorem can we?

 

[matt grime]

Why can't we use the structure theorem for modules over a PID? If there is one, why should you not be able to use it?

As to 'what it would' imply - that is a horrendously open ended question. Do you mean: if a module is torsion must it be finitely generated and/or cyclic?

Why not try to construct some modules? Z is a PID. A modue is an abellian group. A finite abelian group is a torsion module. Are all finite abelian groups cyclic? OF course not. Are all torsion modules finite abelian groups? Of course not.I'm going to refuse to help you again directly until you start to actually play around with some examples. Just look at the *simplest* possible example to see what is going on. Every thread you start will have me posting 'just compute some examples' until you actually give in and do some.

 

[pivoxa15]

Why can't we use the structure theorem for modules over a PID? If there is one, why should you not be able to use it?

I am trying to prove some general results but will be thinking about examples. First though just specifically with the question that if we have a torsion R-module M over a PID R then you are suggesting we can use the structural theorem? However the structural theorem demands M to be finitely generated. A torsion module M does not imply M is finitely generated as you pointed out. So we can't use the strutrual theorem.? This is the issue that led to this thread.

Although I can't seem to find an example of an infinitely generated torsion module.

 

[matt grime]

Im not suggesting using the structure theorem. You're suggesting using it. I don't see why you'd want to do it, since you can *just write down all the examples you need*.

You really can think of a finitely generated torsion module M, right? So what is wrong with M+M+M+... countably many (or even uncountably many) times? + should be taken to mean direct sum. Direct product would also work. If you don't like that then let F be any algebraically closed field of char>0 considered as a Z-module, or even an F_p module, it doesn't matter. This is not finitely generated over Z or F_p. It is torsion. I refuse to believe you couldn't think of an infinite abelian group all of whose elements have finite order which is all you had to do.

 

[pivoxa15]

Oh so an infinitely generated torsion module could just be Z(mod2) + Z(mod2) + ... Z(mod2) + ... infinity...

The annihilator for the whole module is the ideal (2). So it is torsion but infinitely generated.

My question is not really 'is it good to use the structural theorem' but can one use it in the situation to M where only the information 'Given a torsion R-module M over a PID R'. I presume no. As I have just constructed an example of a torsion module that is infinitely generated. The structural theorem is only for infinitely generated modules.

 

[matt grime]

There are structure theorems for infinitely generated modules. But it isn't the one you know.Look at the theorem, look at the proof and decide when you need to invoke the fact that the module is infinitely generated.