Is there any use to $\mathbb{N}$ as a sort of ring mod an infinite number?

Question

I was recently joking with my younger brother regarding rings as subsets of $\mathbb{N}$, and the use of such a construction (don't laugh- we have a weird sense of humor). The discussion got me thinking if it would be of any use to interpret $\mathbb{N}$ as a ring involving infinities akin to viewing, say, $\{0,1,2,3\}$ as a ring $\text{mod }4$. So my question is:

Is there any use to considering $\mathbb{N}$ as a sort of ring of ordinals $(\text{mod }\omega)$ or of cardinals $(\text{mod }\aleph_0)$? If so, what are some examples of situations in which it might be useful to do so? If not, is there any "reason" it's not a very useful way of looking at $\mathbb{N}$?

I debated whether to ask about ordinals or cardinals (my intuition says it should be ordinals, as I believe that's what's more commonly used to extend the natural numbers), but I wanted to be safe, so I asked about both. Feel free to let me know if it's not meaningful to ask my question with regards to one or the other, and I'll be happy to remove that one.

Answer 1

The reason that modulus rings are useful is that they form an isomorphism with $\mathbb{N}$. That is for addition and multiplication $(n * m) \mod b \equiv (n \mod b) * (m \mod b)$. I am not sure you could create an equivalent system between $\mathbb{N}$ and a greater set.

Answer 2

If I had to hazard a guess, my guess would be no, because ordinal addition forms a non-commutative monoid, not an abelian group, so you couldn't really make a ring out of it. While cardinal arithmetic depends on your axioms, in general they too form a monoid for both addition and multiplication.