I want a ring $R$ of "numbers" such that:

For any sequence of congruences $x\equiv a_1 \pmod{n_1}, x\equiv a_2 \pmod{n_2},\dots$ with $a_i\in \mathbb{Z}$ and $n_i\in \mathbb{N}$ such than any finite set of these congruences has a solution $x\in\mathbb{Z}$, there is a $r\in R$ such that $r\equiv a_1 \pmod{n_1}, r\equiv a_2 \pmod{n_2},\dots$

and

For any $r\in R$ and $n\in\mathbb{N}$ there is a $a, 0\leq a< n $ such that $r\equiv a \pmod{n}$.

I think that $S$ has to be the product set of the p-adic integers over all primes p, but what do you call this ring?

(Perhaps there should be a "terminology" tag?)