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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")

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The standard notation is $\widehat{\mathbb{Z}}$. The names I know are "the profinite completion of $\mathbb{Z}$" and "$\mathbb{Z}$-hat".

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As David Speyer says, the most common ways of referring to $\prod_p \mathbb{Z}_p$ are "$\mathbb{Z}$-hat" or "the profinite completion of $\mathbb{Z}$''.

However, I have also heard it called "the Prufer ring", see e.g.

http://mathworld.wolfram.com/PrueferRing.html

I do not endorse this usage, since nowadays "Prufer ring" is a certain kind of commutative ring. An integral domain is a Prufer ring if every finitely generated ideal is invertible. Of course $\widehat{\mathbb{Z}}$ is not a domain. More recently there have been definitions of Prufer ring for non-domains; unfortunately not all of the characterizations of Prufer domains carry over to rings with zero divisors and it is not completely clear to me that there is one standard definition of a Prufer ring. For instance, I have seen that a Prufer ring is a ring in which every finitely generated regular (i.e., containing a non zero-divisor) ideal is invertible, and also that a Prufer ring is a ring in which each finitely generated ideal is flat. (In fact I do not know off-hand whether these two conditions are equivalent! In any case, there are many others...)

For every definition of Prufer ring I have seen, it is at least true that $\widehat{\mathbb{Z}}$ is a Prufer ring.

Also, in Fried and Jarden's authoritative text Field Arithmetic, they refer to $\widehat{\mathbb{Z}}$ as the Prufer group (p. 14 of the third edition). Again, I have heard other things referred to as Prufer groups in the literature...

In summary, "Z-hat" is probably your best bet.

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It was called the Prüfer ring in the old German literature.

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Another (fancy) name is "the free profinite group of rank 1"

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