What happens if I add an $\omega$th digit to the $p$-adic numbers?

Question

If you add $3$-adic numbers like $\dots 111111 + \dots121212 = \dots 010100$ the digits all carry over, so it seems intuitively like you lose a digit at the $\omega$th place, as it's missing an $\omega$th digit to carry over into. So I've been trying to understand what happens when we add an extra digit so that the digit gets incremented by $1$ if there is a cascade of carrying which goes on forever like that.

More precisely, it should get incremented if there is some $k$ so that all digits after the $k$th place carry over. This digit should itself be order $p$, so that you still have an overflow at the $\omega+1$th digit. I'm worried this sounds a bit wishy-washy, as I don't have an explicit way of constructing what I'm talking about, but that's part of why I'm asking.

I think this should be constructible for any ordinal, so that the $p$-adic integer group would be the $\omega$th case of a wider familiy of groups.

This seems like a pretty natural thing to consider- explaining it was quite simple, so one would expect it to have an easy construction, but I can't think of one. the only way of constructing the $p$-adic integer group I know is via constructing the $p$-adic numbers. But I'm not sure how to generalise the construction of the $p$-adic numbers to this, as I'm not even sure it has a ring structure. Not being able to construct it makes it difficult for me to figure out its properties. How to construct this is my main question.

I would also like to know if this has already been studied, and under what name.

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The biggest property I'm interested in is this: is it indecomposable as a group? I'm pretty sure that the $p$-adic integer group is indecomposable (though I tried to look it up and couldn't find anything), and this seems to be built on top of them in a way that wouldn't change that, so I suspect it is, but can't prove it without a construction. I'm interested in this because it would be a very natural way to prove there are arbitrarily large indecomposable Abelian groups, if it can be extended to $p$-adics of any ordinal length.

Other things I'm interested in:

Does it have a ring structure- it's seems like it would, though I can't quite tell if this works without being able to construct it. I think this would be non-commutative, and 'nice' non commutative rings are hard to come by. It seems like it would be local, and a principle ideal ring.

Does it give a division ring in the same way $\mathbb{Z}_p$ gives $\mathbb{Q}_p$?

They seem like they should give totally disconnected topological spaces like the $p$-adic metric space which need arbitrarily large neighbourhood bases.

I would like to hear any other interesting properties about it as well.

Answer 1

There is a simple argument for why your idea shouldn't work at least for $p=2$.

Let $R$ be a topological ring where $\lim_{n\to \infty} 2^n$ converges to some element $\omega \in R$.

$$2 \omega=2\lim_{n\to \infty} 2^n =\lim_{n\to \infty} 2^{n+1}=\omega$$ which gives $\omega=0$.

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For $p$ odd, asuming $\lim_{n\to \infty}p^n=\omega$, we have similarly $p\omega=\omega$. We also have $\omega^2=\omega$ which gives $R\cong R/(\omega) \times R(\omega-1)$.

In $R/(\omega-1)$ we have $\omega=0$.

In $R/(\omega)$ we have $\omega=1$ so $p=1$.

Whence $R$ is the product of a ring where $\omega=0$ and a ring whose characteristic divides $p-1$.

Answer 2

I've been trying to understand what happens when we add an extra digit so that the digit gets incremented by 1 if there is a cascade of carrying which goes on forever like that.

A $p$-adic number does not know that it has been part of a particular type of sum, so it is unclear how you're supposed to decide that you want to put some extra digit after some $p$-adic expansions but not others.

Something else to consider is that a digit carrying is not an intrinsic property of $p$-adic numbers. In $\mathbf Q_3$ we could use digit set $\{0,1,2\}$ or $\{0,4,-1\}$ or $\{0,-1,1\}$. We could also replace the role of $3$ with $6$ or $-12$ (basically $3$ times a $3$-adic unit in $\mathbf Z_3$). Such choices completely change the way digits will be affected under addition.

Example. In the usual expansion for $3$-adic numbers with digits $\{0,1,2\}$ we write $1 = 1$ and $-1 = 2 + 2\cdot 3 + 2\cdot 3^2 + 2 \cdot 3^3 + \cdots$ and their sum $0$ has cascading carrying. But if you use digits $\{0,1,-1\}$ then $1 = 1$, $-1 = -1$, and their sum $0$ has no carrying at all.

So the whole phenomenon of digit carrying under addition is an artifact of the ingredients used to define the expansions: it is not a property of individual $p$-adic numbers and thus I don't think you're going to learn anything interesting about $p$-adic numbers by inserting them into a system with an extra digit at the end.