Is this a helpful way of thinking about modular arithmetic?

Question

Could one consider the structure of $\mathbb{Z_{n}}$ under addition and multiplication to simply be that of the ordinary integers, with one additional axiom - namely that $n=0$? So we would have $\mathbb{Z_{n}}=\{0,1,2,\ldots ,n-1\}$ as usual, for instance.

My question is whether this is a good way of thinking about $\mathbb{Z_{n}}$, or is there a subtlety that I've not considered that breaks it?

Answer 1

Nope, no subtle breakage-this works fine. It's a bit more formal to say the additional axiom is that $n$ is equivalent to $0$ and that you're calculating with representatives of equivalence classes in the resulting relation, but it amounts to the same thing when you get to actually doing modular arithmetic.

Answer 2

I suppose you could start with $n=0$, but there are a lot of consequences that are not at all obvious that you would need to derive. For example, with the integers there are not two, each greater than 1, whose product is 1. But modulo $7$ you have $2\times 4=1$.