Let $R$ be a ring, and $M$ a (left say) $R$ module. For subgroups $M_i$ in $M$, I have been able to prove
$$\left(\sum_{i\in I}M_{i}\right)^{0}=\bigcap_{i\in I}M_{i}^{0}$$ $$\sum_{i\in I}M_{i}^{0}\subset\left(\bigcap_{i\in I}M_{i}\right)^{0}$$
Where $N^0=\left\{n\in N\vert rn=0 \textrm{ for all }r\in R\right\}$ for a subset $N$ of $M$.
Have I got these right, and is it true that the bottom one is just an inclusion, as opposed to an equality, in general? Does this change if the $M_i$ are submodules?
The top one is correct and the bottom one is just an inclusion.
Take two faithful modules $A$ and $B$ (faithful means they have zero annihilator.). Then form $M=A\times B$ and consider $A$ and $B$ as submodules of $M$.
On one hand, $A\cap B=\{0\}$, so it's annihilator is $R$.
On the other hand, the sum of their annihilators is $\{0\}+\{0\}=\{0\}$.