Confusion in notations of number theory

Question

In several materials I saw the notation for complete residue system as follows

$\mathbb{Z}_n=\{0,1,2,3,\cdots, n-1\}$

But in some other materials/ in the same material, It says that the above notation is wrong and has to be represented as $\mathbb{Z}/n\mathbb{Z}$.

Which I have to follow, the reason for my doubt is the notation I used first is used mostly in school materials. If its not correct, than what $\mathbb{Z}_n$ actually mean?

How can I represent complete co-prime residue system? Is it $\mathbb{Z}_n^*$ or $\mathbb{Z}^*/ n?$

And finally how can I interpret $\mathbb{Z}/n\mathbb{Z}$? Is it $\mathbb{Z} - n\mathbb{Z}$ . If it is, then what $n\mathbb{Z}$ mean?

Answer 1

The notation mostly depends upon whether the author of the text is planning on dealing with $p$-adic numbers or not. If the author is not planning on discussing them (as is the case in many elementary number theory or group theory texts), then the notation $\mathbb{Z}_n$ seems to be fairly common. When $p$-adic numbers are around, however, that notation generally means something different (which I won't discuss here for fear of being off topic, unless you leave a comment below asking for a description).

The other notation (and what probably many number theorists would consider to be the "better" notation) is to use $\mathbb{Z}/n\mathbb{Z}$. This notation implies that we are looking at the numbers modulo n, and the reason we do this has to do with some mathematical structures called "quotient rings." $n\mathbb{Z}$ means all integer multiples of the number $n$. $\mathbb{Z}/n\mathbb{Z}$ means that we are declaring all multiples of $n$ to be equivalent, which is another way of saying that we are taking the numbers modulo $n$.

You would probably want to represent the co-primes as $(\mathbb{Z}/n\mathbb{Z})^*$.

Answer 2

$\mathbb{Z}_n$ is the group of integers, modulo $n$, under addition. This is what we usually use when we use modular arithmetic and residue classes, etc. So $\mathbb{Z}_n = \{\overline{0}, \overline{1}, \overline{2}, ... \overline{n-1}\}$.

Now $\mathbb{Z}/n\mathbb{Z}$ is an example of a quotient group. $\mathbb{Z}/n\mathbb{Z} = \{0+n\mathbb{Z}, 1+n\mathbb{Z},...,(n-1)+n\mathbb{Z}\}$, which can be naturally identified with $\mathbb{Z}_n$.

I'd assume the group of coprimes would be denoted $(\mathbb{Z}/n\mathbb{Z})^*$