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.)

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) $\endgroup$ – Wojowu Jan 1 '16 at 13:04