Hi, apologies for the late reply. The last few days have been a mess so no time for math

The only thing I can come up with is that since $\displaystyle M \cap N \subset M+N $, then $\displaystyle (M+N)^0 \subset (M \cap N)^0$, but that doesn't really prove either (1) nor (2)..

Indeed, that doesn't...but $\displaystyle M\cap N\subset M\Longrightarrow M^0\subset (M\cap N)^0\,,\,\,and\,\,\,also\,\,\,M\cap N\subset N\Longrightarrow N^0\subset (M\cap N)^0$ and now you get what you wanted...
I figure that I need to show that $\displaystyle M^0 \cap N^0 \subset (M^0+N^0)$. If I assume that (2) is true, I should be able to take the annihilator of both sides of (2) and get to the wanted result.

$\displaystyle ((M+N)^0)^0 \subset (M^0 \cap N^0)^0 $

Then by (a) I have that

$\displaystyle (M^0 \cap N^0) \subset (M+N)^0$

Ah, I find this to be really hard!

Yes, it really requires concentration. Try again, since the last days you did no much math (or at all). Tonio
Thanks!