Could we say any $k \in \mathbb{Z}$ also in $\mathbb{Z}/n\mathbb{Z}$?

Question

As a student currently taking abstract algebra / introductory group theory, one thing that I wanted to clear up is exactly how we define $Z_n$. Before I learned about quotient groups, I was taught that $\mathbb{Z}_n$ contained all integers $k \in \{0,1,\dots, n-1\}$. I was also taught this addition in this group was defined as $a+b \mod n$.

However, what wasn't made clear to me was if the elements in the above set were representative elements of the equivalence classes, or if in the realm of $\mathbb{Z}_n$, integers greater than $n$ and less than $0$ simply didn't exist. Looking at this wiki, it's difficult for me to tell if the elements are representative elements.

When I learned we can write $\mathbb{Z}_n$ as $\mathbb{Z}/n\mathbb{Z}$, the quotient group, it made me lean towards thinking that any integer in $\mathbb{Z}$ is also in $\mathbb{Z}/n\mathbb{Z}$, since elements of $\mathbb{Z}/n\mathbb{Z}$ are cosets.

So I want to clarify—is it completely acceptable to say, for example, that $(5+6) \in \mathbb{Z}_4$? When writing homomorphisms $\phi:\mathbb{Z}_4 \rightarrow \mathbb{Z}_n$ this comes up since I may want to write $\phi(2+2) = \phi(4) = \phi(0)$ but I was never sure if that was acceptable or if I could work with integers greater than $n$ in $\mathbb{Z}_n$.

Answer 1

Elements of $\mathbb{Z}/n\mathbb{N}$ are equivalence classes. Since writing $\phi([2]_4+[2]_4)$ or $\phi([2+2]_4)$ gets annoying pretty fast, you usually implicitly intend that the standard notation for a number, like $137$, denotes its equivalence class. Therefore with this understanding it's acceptable to write $(5+6)\in\mathbb{Z}_4$.

Answer 2

Technically one can consider the elements of $\mathbb Z_n$ to be either $\{0,1,\ldots,n-1\}$ or equivalence classes modulo $n$. Each of these choices works well enough, but the second of them generalizes better to quotients in abstract algebra, so that is what textbooks most often do.

Neither of these conventions make, for example, the number $5$ an element of $\mathbb Z_4$.

However, it is not uncommon in calculations to nevertheless pretend that it is. If pressed for specifics, we can justify it by saying that there is an "invisible homomorphism" $\mathbb Z\to\mathbb Z_n$ being applied at each point in your formulas where it is necessary for things to make sense.

In any case, the real point you need to worry about is not what things really are, but whether you're explaining things in a way such that it is clear to both you and your readers what you mean, and in particular when you're working with actual integers and when you're speaking about modular arithmetic. You can get away with quite a bit of vagueness when as long as you're talking about addition, subtraction and multiplication (because these operations commute with the invisible homomorphism), but if you begin speaking about division or multiplicative inverses, being explicit becomes much more important.

Answer 3

A standard construction of $\mathbb{Z}_n$ without group/ring theory is via equivalence classes of the relation

$$ a \equiv b \pmod{n} \iff n | (a-b) $$

in which case we note $[a]$ for the class of $a \in \mathbb{Z}$. Now, if $r_n$ is the 'remainder in the division by $n$' function, an immediate consequence is that

$$ a \equiv b \pmod{n} \iff r_n(a) = r_n(b). $$

Since for any integers $a,b \in \mathbb{Z}$ we have that $$ r_n(a+b) = r_n(r_n(a)+r_n(b)) \ \text{ and } \ r_n(ab) = r_n(r_n(a) \cdot r_n(b)), \tag{1} $$ this implies that the operations $[a] + [b] := [a+b]$ and $[a][b] := [ab]$ are well defined because they are independent of the representatives chosen, and one can verify that this gives $\mathbb{Z}_n$ a ring structure (without ever saying the word ring). Since by $(1)$ we get that

$$ [a+b] = [[a]+[b]] \text{ and } [ab] = [[a][b]], \tag{2} $$

we often abuse notation (and this fact) by doing the operations in the integers or taking remainder in the order we please, because $(2)$ says that these 'commute', in some sense. Thus, we drop the class symbol altogether, which may cause some confusion, if one is unaware of the former convention.

On the other hand, provided that you know some ring theory, an equivalent definition of $\mathbb{Z}_n$ is that of $\mathbb{Z}/n\mathbb{Z}$ with the canonical ring structure of a quotient. The same relations as in $(2)$ hold, hence giving rise to the same abuse of notation.