Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it? Fix a positive integer $p$, possibly prime.
For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow \mathbb{Z}/p^n \mathbb{Z}.$$ For any given integer $k$, we can think of $\pi_n(k)$ as the remainder when $k$ is divided by $p^n$.
Now clearly, $\pi_n$ will never be injective. If we wanted to get around this, we could trying bundling up all the $\mathbb{Z}/p^n \mathbb{Z}$ together as a direct product. So define:
$$\mathbb{Z}/p^\mathbb{N} \mathbb{Z} = \prod_{n:\mathbb{N}} \mathbb{Z}/p^n\mathbb{Z}$$
There is, of course, a unique ring homomorphism $\pi : \mathbb{Z} \rightarrow \mathbb{Z}/p^\mathbb{N} \mathbb{Z}.$ In fact, it can be computed pointwise: $$(\pi(k))_{n} = \pi_n(k)$$
In some sense, we might say that $\pi(k)$ is the sequence of remainders obtained by attempting to divide $k$ by increasingly large powers of $p$.
The cool thing about this is that:


*

*I'm pretty sure that $\pi$ is always injective; hence, we've included the integers in a bigger number system. Perhaps there are situations where we're working away in $\mathbb{Z}$ but we find that there aren't "enough" integers to do what we're trying to do; if so, one idea would be to look in $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ for an appropriate choice of $p.$

*Suppose $p$ is a prime number. Then to compute the number of copies of $p$ in the prime factorization of an integer $k$, we can simply count the number of $0$'s in the sequence $\pi(k)$. Furthermore, these $0$'s will always occur as an initial segment.

Question. Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, is it associated with a standard name or notation, and where can I learn more about it?

 A: $\newcommand\Z{\mathbf{Z}}$
Call the ring $R$.
You will notice that the image of $\Z$ in $R$ under the map $\phi:\Z \rightarrow R$ you constructed has the property that, if you write is as:
$$\phi(a) = (a_1,a_2,a_3, \ldots )$$
then:


*

*$a_2 \in \Z/p^2$ has remainder $a_1$ when reduced modulo $p$, 

*$a_3 \in \Z/p^3$ has remainder $a_2$ when reduced modulo $p^2$, 

*$a_4 \in \Z/p^4$ has remainder $a_3$ when reduced modulo $p^3$, 


and so on...
So you might consider replacing your ring $R$ by the subring $A \subset R$ consisting of (infinite) sequences which satisfy this property (namely, the ring consisting of sequences such that that each element in $\Z/p^{n+1}$ reduces modulo $p^n$ to the corresponding element in $\Z/p^n$ term). If you do that, then the corresponding ring $A$ is exactly the $p$-adic numbers $\Z_p$. The ring $A$ has all the  nice properties you noted about $R$, but (in addition) has much nicer ring theoretic properties: it's torsion free, it's reduced and irreducible (so an integral domain), it's a complete local ring, it has a natural topology, and finally, the image of $\Z$ under the map $\phi$ is dense with respect to this topology. (Equivalently, for any element $a \in A$, there is a $x_n \in \Z$ so that $\phi(x_n)$ and $a$ agree to the first $n$ terms.)
Naturally, the ring of $p$-adic numbers $\mathbf{Z}_p$ is indeed widely studied and very useful. (If you insist on considering $R$ instead, then I don't imagine it has a name or is particularly studied.)
