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 $R$ 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? **Edit:** It already exists but it is called "names")



What do you call this ring?
The standard notation is $\widehat{\mathbb{Z}}$. The names I know are "the profinite completion of $\mathbb{Z}$" and "$\mathbb{Z}$-hat".


