Edit: Since there has been a recent resurgence in activity, I may as well share what I have found. This construction for prime base p is seemingly equivalent to what is called a p-adic solenoid in the literature. I haven't still been able to find whether it's possible to define a well-behaved multiplication in these objects to give them the structure of a ring, but this should provide some context and hopefully help answer the question.
Summary of the question:
Doubly-infinite decimal numbers are "numbers" whose decimal expansions are allowed to extend infinitely both to the left and right of the decimal point, for example
$$\ldots8906253.141592\ldots$$
with some identifications between decimals that are equivalent in a sense (see below). They can be added and subtracted without difficulty using carrying arithmetic, but one has trouble multiplying them with the usual algorithm. Is there any way to define multiplication of these numbers in a consistent way?
Here is a possible formal construction of doubly-infinite decimals. Let $D_n$ be the set of formal Laurent series
$$\sum_{k=-\infty}^{\infty} a_k x^k$$
with coefficients in $\mathbb{Z}/n\mathbb{Z}$. The variable $x$ is to be thought of as the number $n$ itself. The elements of $D_n$ can be represented in a way analogous to decimal numbers in base-n, where we now allow digits to extend infinitely to the left of the decimal point. This set contains, among others, the decimal representations of both real and n-adic numbers (not the numbers themselves, because we don't impose a metric structure nor a notion of convergence in $D_n$, we are treating those decimals purely formally).
Given two elements in $D_n$, we can define their sum by extension of the standard carrying algorithm (note that this is different from the term-by-term addition of formal Laurent series). The sum can be proved to be (up to the equivalences below) associative, commutative, and with an identity (...00.00...). We now define the equivalence relation $a \sim b$ iff $a = b+h$, where $h$ has a decimal representation of the form
$$...(abc...d)(abc...d).(abc...d)(abc...d)...$$
i.e., an infinite repeating decimal in both directions. The motivation is that those numbers have the property that they stay the same if we shift the decimal point a certain number of times to the right, i.e. if they were proper base-n numbers they would satisfy $n^m h = h$ for some $m$, so they can be considered alternate representations of zero. We also identify decimals of the form $...001.000... \sim ...000.999...$ (9 stands for the last digit in base-n) since they behave the same way with respect to addition. The set of doubly-infinite decimal numbers is the quotient set
$$\mathbb{D}_n = D_n/\sim$$
We can now define additive inverses by
$$- b = 0 - b = (\ldots999.999\ldots) - b$$
thanks to the equivalence relation above, in analogy to one's complement arithmetic. With this, $\mathbb{D}_n$ becomes an abelian group with respect to addition.
What about multiplication? My intuition tells me that it's not possible, because addition of decimal numbers behaves locally with respect to the digit sequences (localized perturbations in the digit sequences of the summands correspond to localized perturbations in the result; compare the decimal expansions of $e+\pi$ and $(10+e)+\pi$), so if we perform addition digit by digit one can clearly see what number it is "converging" into, while multiplication of decimals has global effects that spoil the digit convergence (compare $e\times\pi$ with $(10+e)\times\pi$). This is probably related to how it's impossible to multiply formal Laurent series unless you only allow finitely many terms of negative degree. But I don't know whether this argument rules out the existence of multiplication completely, or just the possibility of performing it constructively by the usual algorithms.
Ideally one would like multiplication of doubly-infinite decimals to reduce to the usual product of real and n-adic numbers when $\mathbb{D}_n$ is restricted to $\mathbb{R}$ and $\mathbb{Q}_n$ respectively, and share the properties of commutativity, associativity, distributivity with respect to addition and existence of an identity. How can I prove whether it exists or not?
