1. ## Show an isomorphism

I have to find this isomorphism:

$\displaystyle (P_{\alpha +1}\oplus Q_{\alpha +1})/(P_\alpha \oplus Q_\alpha) \simeq P_{\alpha +1}/P_\alpha \oplus Q_{\alpha +1}/Q_\alpha$

Where $\displaystyle P_\alpha = S_\alpha \cap P$ and $\displaystyle Q_\alpha = S_\alpha \cap Q$ and $\displaystyle {S_\alpha}$ is a well ordered increasing sequence of sub modules for $\displaystyle M=P\oplus Q$.

I know that it is something about sending the residue class $\displaystyle (\overline{P_{\alpha +1}, Q_{\alpha +1}})$ into the pair of residue classes $\displaystyle (\overline{P_{\alpha +1}}, \overline{Q_{\alpha +1}})$

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

Any help would be nice, thanks.

2. Originally Posted by Carl
I have to find this isomorphism:

$\displaystyle (P_{\alpha +1}\oplus Q_{\alpha +1})/(P_\alpha \oplus Q_\alpha) \simeq P_{\alpha +1}/P_\alpha \oplus Q_{\alpha +1}/Q_\alpha$

Where $\displaystyle P_\alpha = S_\alpha \cap P$ and $\displaystyle Q_\alpha = S_\alpha \cap Q$ and $\displaystyle {S_\alpha}$ is a well ordered increasing sequence of sub modules for $\displaystyle M=P\oplus Q$.

What do you mean by "well-ordered increasing seq. of submod. of..."?? That is indexed by a countable index set and thus $\displaystyle S_1\subset S_2\subset ....\subset S_n\subset\ldots$ , or

only that for any two indexes $\displaystyle \alpha,\,\beta$ , either $\displaystyle S_\alpha\subset S_\beta\,\,\,or\,\,\,S_\beta\subset S_\alpha$ ?

Tonio

I know that it is something about sending the residue class $\displaystyle (\overline{P_{\alpha +1}, Q_{\alpha +1}})$ into the pair of residue classes $\displaystyle (\overline{P_{\alpha +1}}, \overline{Q_{\alpha +1}})$

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

Any help would be nice, thanks.
.

3. Pick any $\displaystyle (x,y)\in P_{\alpha+1}\oplus Q_{\alpha+1}$, then your map is just $\displaystyle (\overline{x,y})\mapsto(\bar{x},\bar{y})$. You need to show that 1) the map is well-defined, by showing if (x1,y1), (x2,y2) differ by something in $\displaystyle P_\alpha\oplus Q_\alpha$, 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.