Thread: Show an isomorphism

I have to find this isomorphism:



Where and and is a well ordered increasing sequence of sub modules for .

I know that it is something about sending the residue class into the pair of residue classes

But how to argue that this is a homomorphism?
And what would the kernel be?

Any help would be nice, thanks.

Originally Posted by Carl

I have to find this isomorphism:



Where and and is a well ordered increasing sequence of sub modules for .


What do you mean by "well-ordered increasing seq. of submod. of..."?? That is indexed by a countable index set and thus , or

only that for any two indexes , either ?

Tonio


I know that it is something about sending the residue class into the pair of residue classes

But how to argue that this is a homomorphism?
And what would the kernel be?

Any help would be nice, thanks.

.

Pick any , then your map is just . You need to show that 1) the map is well-defined, by showing if (x1,y1), (x2,y2) differ by something in , then they map to the same thing, and 2) is a homomorphism (additive property and scalar multiplication property), which is easy to prove, and 3) 1-1 and onto, which is also not hard.