"Prime decomposition" for the profinite integers, adeles, or p-adics? The strictly positive integers can be decomposed into a product of prime powers. Likewise, the positive rationals can be decomposed into a product of (possibly negative) prime powers.
Another way to state this is that the underlying multiplicative monoids are free and admit the primes as a basis.
If negative integers or rationals are allowed, then we no longer get a free monoid. But, we can still get a suitable prime decomposition if we create an additional "basis" element, representing -1, that takes exponents in $\Bbb Z/ 2 \Bbb Z$.
I'm wondering if a similar sort of decomposition exists for any of the more interesting structures thrown around in number theory, namely


*

*The profinite integers, and/or its group of units

*The ring of adeles, and/or its group of units

*The field of p-adic numbers for some p


I'm particularly interested in the case where the "basis" is countable, like the primes themselves.
Now, since these are all uncountable, it's clear that a decomposition into a direct sum of countable groups won't work. But perhaps we can allow infinitely many non-zero exponents, or allow the exponents to take values in some interesting group. Does anything like this work?
The supernatural numbers seem related here, but I can't tell exactly what the relationship is, or if they embed into one of the groups mentioned (or vice versa).
 A: There is a notion of unique factorization domain, which is a commutative integral domain whose nonzero elements admit unique prime factorizations up to units. (If we don't ignore units then there's no hope of getting uniqueness.)
The $p$-adic integers $\mathbb{Z}_p$ have this property mostly by virtue of having a lot of units: every $p$-adic integer is uniquely of the form $p^n$ times a unit. In fact $p$ is the only prime in the $p$-adics! This reflects the fact that $\mathbb{Z}_p$ is a local ring. 
The group of units is well understood. Let me assume that $p$ is odd because something annoying happens when $p = 2$. The group of units is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{\times}$ times the subgroup of units congruent to $1 \bmod p$. This subgroup is in turn isomorphic to the additive group $\mathbb{Z}_p$, with the isomorphism given by the exponential
$$\mathbb{Z}_p \ni a \mapsto (1 + p)^a \in \mathbb{Z}_p^{\times}.$$
In particular, it is "topologically generated" by $1 + p$.
The profinite integers and the adeles don't fall into this framework because they are not integral domains. In a ring that's not an integral domain it's very unclear what one ought to mean by unique prime factorization of a zero divisor. Nevertheless, since the multiplicative monoid and group of units functors both send products to products, once you've understood what's going on for $\mathbb{Z}_p$ you more or less understand what's going on in these more complicated cases. 
