Another reference to a proof of Specker's theorem is [Zagier's St Andrews problems][1].

**Added**
Also rings such as $\mathbb{Z}$ with this property are called
[slender rings][2].


  [1]: http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html
  [2]: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-49/issue-2/Slender-rings-and-modules/pjm/1102945104.full