Sure, this has been studied, and various generalizations are known (infinite
sets, three and more set summands, abelian groups other than the group of
integers, asymptotic decompositions). Ostmann's conjecture dating back to
1968 states that the set of prime numbers is asymptotically additively
indecomposable, and this is still open. You can check, say, the paper of
Elsholtz and Harper [Additive decompositions of sets with restricted prime
factors][4] and the
references therein for a pretty comprehensive overview. An open conjecture of
Sarkozy is that the set of quadratic residues modulo a prime is
indecomposible. (There is a nice argument of Shkredov showing that the set of quadratic residues cannot be represented as $A+A$, but it is not known whether it can be represented, say, as $A-A$; see [this paper][1].) Two more links: [Large sets in finite fields are sumsets][2] by Alon, and Problem 4.11 [here][3].

  [1]: http://math.haifa.ac.il/seva/Papers/QRDiff.pdf
  [2]: http://www.cs.tau.ac.il/~nogaa/PDFS/sumset.pdf
  [3]: http://math.haifa.ac.il/seva/Papers/montpr.dvi
  [4]: https://www-ams-org.ezproxy.haifa.ac.il/journals/tran/2015-367-10/S0002-9947-2014-06384-8/S0002-9947-2014-06384-8.pdf