Notation in $p$-adic integers

Question

When we write $\mathbb{Z}_3$, does it mean $\mathbb{Z}/3\mathbb{Z}$? Also, does $3\mathbb{Z}_3$ mean $0 \pmod 3$?

Answer 1

Many notations in mathematics are overloaded. Whenever you are working in mathematics, it is very important to keep track of the context in which you are working. For example, in the expression $$ 3^s = 2 \implies s \in \left\{ \log_{3}(2) + i\frac{2\pi}{\log(3)} \right\},$$ they symbol $i$ (clearly?) represents the imaginary unit, i.e. the complex number which, when squared, gives $-1$. On the other hand, in the notation $$ \sum_{i=0}^{\infty} \alpha^i = \frac{1}{1-\alpha}, $$ the variable $i$ is the index of summation. In both cases, the notation should be unambiguous, as the context makes it clear what is going on.

Similarly, the notation $\mathbb{Z}_3$ is potentially ambiguous, but is typically clear with the correct context. In the context of anything $p$-adic, the notation $\mathbb{Z}_3$ unambiguously represents the $3$-adic integers. If you are working in a $p$-adic setting (or, more generally, if the words "local field" appear in your work), then it is safe to assume that $\mathbb{Z}_3$ is the set of $3$-adic integers. On the other hand, it is quite common to use $\mathbb{Z}_3$ to denote the quotient group $\mathbb{Z}/3\mathbb{Z}$ in other contexts. If you are not explicitly working in a setting where the $3$-adic integers might pop up, it is safe to assume that $\mathbb{Z}_3 = \mathbb{Z}/3\mathbb{Z}$. If you fear that there is any potential for ambiguity, then make it clear—unambiguously define your notation.

With respect to the last part of your question, $3\mathbb{Z}_3$ almost certainly denotes the set $$ 3\mathbb{Z}_3 = \{ 3x \mid x \in \mathbb{Z}_3 \}, $$ which is the ball of radius $\frac{1}{3}$ centered at zero in the space of $3$-adic numbers.

Answer 2

$\Bbb Z_n$ is a bit of an ambiguous notation without further context. I've seen it used both as a means of denoting the integers mod $n$, and as a means of denoting the $n$-adic integers - two very different things as you might imagine.

$\Bbb Z / n \Bbb Z$ however is less ambiguous from my experience: it denotes the quotient set of the integers mod the set of the integer multiples of $n$, i.e. it is the integers mod $n$. I've never seen it used to denote a $p$-adic entity.

So in a $p$-adic context you'll probably see the subscript notation when speaking of the $p$-adic integers and to avoid confusion in said context you should see the quotient set notation whenever modular arithmetic and such is relevant.