I am missing something that I imagine is glaringly obvious with the primary decomposition theorem applied to a finitely generated torsion module over a PID. From my reading of the theorem, the following should be true, but is obviously incorrect. What am I not seeing?
Take $\mathbb{Z_6}$ as the finitely generated torsion module over $\mathbb{Z}$. This has order $\mu = 6 = p_1*p_2=2*3$, where $2$ and $3$ are distinct nonassociate primes in $\mathbb{Z}$. Then $\mathbb{Z_6} = (\mu$/$p_1)$$\mathbb{Z_6}$ $\oplus$ $(\mu$/$p_2)$$\mathbb{Z_6}$ = $3\mathbb{Z_6}$ $\oplus$ $2\mathbb{Z_6}$.
This is obviously not correct.
The relevant part of the theorem, stated from Roman:
Let $M$ be a (finitely generated) torsion module over a PID $R$, with order $\mu = p_1^{e_1}...p_n^{e_n}$ where the $p_i$'s are distinct nonassociate primes in $R$. Then $M = M_{p_i}$ $\oplus$ ... $\oplus$ $M_{p_n}$ where $$M_{p_i} = (\mu/{p_i}^{e_i})M = \{v \in M \mid p_i^{e_i}v = 0\}.$$
Clarification would be appreciated. Sorry if this is completely trivial.
Actually, that really was a bad question. After thinking a little bit more, it looks like my confusion involved the definition of a direct sum more than the theorem itself. The example obviously checks out as stated. Carry on.