Denote $p$ a prime number and $\mathbb Z \_p$ the ring of $p$-adic integers.
We have a canonical injective ring homomorphism $:\mathbb Z \rightarrow \mathbb Z\_p$ for all $p$. But $\mathbb Z$ is not the largest ring that maps into all $\mathbb Z\_p$. 
Consider for instance the ring of all formal series 
$$
F := \left\{ \sum\_{n=0}^\infty a\_n \cdot n! \; ,\; 0\leq a\_n\leq n \right\}.
$$
Since $(n!)\_n$ converges $p$-adically to zero for every $p$, this gives a canonical morphism $\phi\_p: F\rightarrow \mathbb Z\_p$. 


I must admit that $\phi\_p$ is not injective, but the intersection 
$\bigcap\limits\_p \ker \phi\_p$ is zero. 
So, what is known about rings $R$ containing $\mathbb{Z}$ and having "canonical" morphisms $\phi\_p:R\rightarrow \mathbb Z\_p$ with $\bigcap\limits\_p \ker \phi\_p=0$?


To start with, if $x\in R$ is an element that satisfies a polynomial equation 
with coefficients in $\mathbb Z$, then $x\in \mathbb Z$ (by Chinese remainder 
theorem). So $R$ must be a transcendental extension of $\mathbb Z$.


Is there a maximal ring resp. a good notion of "maximal element" of such rings? (For example, a ring $R'$ with an injection $:R\rightarrow R'$ making all $\phi\_p$-diagrams commutative.) 



Rings that inject in all p-adic integers
The ring you are looking for is $\widehat{\mathbb{Z}}={\displaystyle\lim\_{\leftarrow}\mathbb{Z}/N\mathbb{Z}}$.


This has canonical maps to $\mathbb{Z}\_p={\displaystyle\lim\_{\leftarrow}\mathbb{Z}/p^k\mathbb{Z}}$ induced by taking $\mathbb{Z}/N\mathbb{Z}\twoheadrightarrow \mathbb{Z}/p^k\mathbb{Z}$ (where $p^k$ is the biggest power of $p$ dividing $N$).


Your condition asks that the product map $\widehat{\mathbb{Z}}\to \prod\_p \mathbb{Z}\_p$ should be injective; in fact, by the Chinese Remainder Theorem, this map is an isomorphism: $\widehat{\mathbb{Z}}\cong \prod\_p \mathbb{Z}\_p$. So this is the universal such ring.




---


A broader perspective: passing from $\mathbb{Z}$ to $\widehat{\mathbb{Z}}$ amounts to taking the completion of $\mathbb{Z}$ w.r.t the topology induced by the inclusion $\mathbb{Z}\hookrightarrow \prod\_p \mathbb{Z}\_p$, so the ring $\widehat{\mathbb{Z}}$ we obtain is the *closure* of $\mathbb{Z}$ inside $\prod\_p \mathbb{Z}\_p$. The fact that this is *all* of $\prod\_p \mathbb{Z}\_p$ means that $\mathbb{Z}$ is **dense** in $\prod\_p \mathbb{Z}\_p$. This density is a weak form of "approximation" (it's the Chinese Remainder Theorem), which is significantly generalized by the classical theorems of [weak and strong approximation](https://en.wikipedia.org/wiki/Approximation_in_algebraic_groups) (and even [superstrong approximation](https://en.wikipedia.org/wiki/Superstrong_approximation)).


