What does $\mathbb{Z}_3$ mean?

Question

Can somebody tell me what $\mathbb{Z}_3$ means?

Does it mean the positive integers up to $3$?

$\mathbb{Z}_3\stackrel{?}{=}\{1,2,3\}$.

Or is it something else?

Answer 1

It may mean either the field with $3$ elements $\mathbf Z/3\mathbf Z$ or the ring of $3$-adic integers, i.e. $\;\biggl\{\sum\limits_{k\ge0}a_k 3^k\:\Big \vert\: \forall k,\;0\le a_k\le 2\biggr\}$.