Why does finite generation of a semisimple module imply finitely many summands?

[peteryellow]

Can somebody help me with the following proof:

Let M be a semisimple module, say M = +_IS_i, where + denotes direct sum and S_i is a simple module.
Then the number of summands is finite if and only of M is finitely generated.

I have problem with understanding the proof of the following in my notes:

if M is finitely generated then the number of summands is finite

Can somebody help me in this argument.

 

[morphism]

This follows immediately from the definition of a "finitely generated" module.

 

[peteryellow]

I know that it is quite clear but still there is an argument which I don't understand and I somebody can help me with this, I will be greatful.

 

[morphism]

What argument, exactly?

 

[mathwonk]

try proving the contrapositive, that if M is any non zero module, that an infinite direct sum of copies of M cannot be finitely generated.

recall the definition of direct sum, and in particular that only a finite number of summands can occur in each element of a direct sum.

 

[peteryellow]

Ok the argument for this theorem in my notes which I don't understand is:

Let M be finitely generated by u_1,...,u_r say. For each u_j we can find finitely many terms S_i whose sum contains u_j. Hence all the u_j are contained in the sum of a finite subfamily of the S_i and this family generates M so that I must be finite.

I don't understand details of this so it will be good if you can help me with the details. Thanks.

 

[peteryellow]

mathwonk, I have a proof of this which I don't understand.

 

[morphism]

Ok the argument for this theorem in my notes which I don't understand is:

Let M be finitely generated by u_1,...,u_r say. For each u_j we can find finitely many terms S_i whose sum contains u_j. Hence all the u_j are contained in the sum of a finite subfamily of the S_i and this family generates M so that I must be finite.

I don't understand details of this so it will be good if you can help me with the details. Thanks.

If you understand the appropriate definitions ("direct sum" and "finitely generated"), then the details will be crystal clear.

 

[peteryellow]

Nut why is it true that this family generates M

 

[morphism]

It generates it in the sense that its sum is M. (And this is true because M is generated, as a module, by u_1, ..., u_r.)

 

[peteryellow]

Thanks a lot