I have two questions about modulo n:

1). In my notes I have $\displaystyle \mathbb{Z}_n$ quoted everywhere! My big problem is: how do I know if this is addition modulo n or multiplication modulo n?

2). My second question is that I have this:

$\displaystyle \mathbb{Z}_n \oplus \mathbb{Z}_m$.

What does this mean? I know it's the direct sum, so does it mean this:

For example, $\displaystyle \mathbb{Z}_3 \oplus \mathbb{Z}_4=\{0,1,2 \} \oplus \{0,1,2,3 \}=\{0,2,4,3\}$?

(I just went through and added the 1st number of each set to the 1st number of the other (and so on)). My big problem was that I didn't have anything to add to the 3 =S

If someone can clarify these it would help a lot!