If we say "classes of non-zero integers modulo $n$", why does this not include the $0$ class? I suppose this is a bit of a wording question more than anything else - I'm working through group theory and was learning that "the (classes of) non-zero integers modulo $p$ form an Abelian group under multiplication." It's the wording of this that gets me a bit confused. Let's say $p = 3$. From my understanding the group described is meant to contain $\{[1],[2]\}$. I completely get why this would be an Abelian group under multiplication. However, if we're looking at all the non-zero integers modulo $3$, it seems to me like the "non-zero" attribute binds to the integer part only. So it's $\mathbb{Z} - \{0\}$ (which would include $3,-3,6,-6,...$) modulo $3$. Hence, since integer multiples of 3 are included in this set, $[0]$ would also be included. But clearly my interpretation can't be the case since including $[0]$ would mean it's not a multiplicative group. 
In summary, it's like I'm having trouble with the order of operations here:
Interpretation 1: (set of non-zero integers) modulo $p$
Interpretation 2: set of non-zero (integers modulo $p$)
So am I supposed to interpret it like interpretation 2? Since it seems like interpretation 1 would include $[0]$.  

I think the "blocking" isn't and shouldn't be "[non-zero integers] [modulo $p$]" but rather "[non-zero][integers modulo $p$]"
You have three [integers modulo $3$].  They are $[53], [-216]$ and $[3{,}691]$.  $[53]$ and $[3{,}691]$ are non-zero [integers modulo $3$].  And $[-216]$ is not a non-zero [integer modulo $3$].
